Global wind patterns

WEBVTT
Kind: captions
Language: en

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- [Instructor] Today, we're going to talk

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about Global Wind Patterns.

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Wind determines more than just
the best places to fly kite.

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Global wind patterns help
control where it rains.

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What kinds of species
can survive in an area

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and even where tropical rainforests
and deserts are located.

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In other words, global wind patterns

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are really important to life.

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And one of the reasons we have
global wind patterns at all,

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is actually because of the sun.

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Sunlight shines on the earth like this.

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As you can see, the sunlight
hits the equator directly

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but, the light hits the North
and South poles at an angle

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kind of skimming the surface like this.

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So the equator is getting direct sunlight

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and the poles are only
getting indirect sunlight.

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With all the direct sunlight,

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the air on the equator gets really hot.

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And the air around the poles
doesn't heat up as much

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because it's only getting
indirect sunlight.

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And you may be thinking,

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what does this all have
to do with airflow?

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Let's take a closer look at the equator

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to see what's happening.

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So imagine you're standing
on a piece of land

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near the equator.

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When the direct sunlight hits the equator,

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the hot air near the
ground begins to rise up

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because hot air rises.

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All the direct sunlight also
causes evaporation to increase,

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which means that this air
that's rising up right here

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is both hot and moist.

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But once all this hot moist air reaches

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a high enough altitude,

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it begins to expand and cool down.

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The water vapor in the air
begins to condense into clouds

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and it eventually falls as
rain around the equator.

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The air which is now cool
and holds less moisture,

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sinks down to the ground
because cold air sinks

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and the cycle repeats

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with the hot moist air rising
and the cool dry air falling.

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And the fact that the hot air rises up,

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it means that this area right here

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is an area of low pressure.

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Because the air is rising up,

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it creates a space for cooler air

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in surrounding areas to
move in and take its place.

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And over here, where the
cool air is coming down

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to the ground, that's
an area of high pressure

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because all of that cool
dry air is coming down

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and pushing the air below it away.

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This cyclical movement
of air create something

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called a convection cell.

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If the earth wasn't spinning,

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we would just have one convection
cell in each hemisphere

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where the air would
heat up at the equator,

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move up towards the poles and sink down.

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And in the 18th century,

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this was how some scientists

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believe global wind patterns worked.

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But, because the earth is spinning,

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the earth's rotation pushes
air masses from East to West.

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This movement of air
creates a clockwise pattern

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in the Northern hemisphere and
a counter-clockwise pattern

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in the Southern hemisphere.

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This is called the Coriolis effect.

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And this movement of air
because of the earth's spin,

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causes us to actually get
three convection cells

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in each hemisphere.

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These two, the two closest to the equator

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are called the hadley cells.

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They're between the equator

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and the 30 degree latitude
marks in both hemispheres.

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These next two are called the ferrel cells

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or the temperate cells.

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And these are located between
a 30 and 60 degree marks

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in both the Northern and
Southern hemispheres.

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And lastly, we have the polar cells

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which as you can probably
guess are right by the poles.

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So we have polar cells
up here at the North pole

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and we also have polar cells
down here at the South pole.

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And these convection cells
create prevailing winds

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that move heat and
moisture around the earth.

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Let's take a look at what
happens in the bottom half

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of each convection cell,

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the parts closer to the ground.

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Because these parts are closer to us,

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we experience the air movement as wind.

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So on the bottom of this
convection cell, the hadley cells,

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the cold air is moving
towards the equator.

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So that means that the prevailing winds

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would also be moving towards the equator.

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Winds are named after
where they come from.

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So, these two winds are
called the Northeast

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and Southeast trade winds

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because they come from the
Northeast and Southeast

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and they move West.

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And the bottom halves of
the ferrel convection cells,

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take cool air from a 30 degree line

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and pull it towards the
60 degree latitude line.

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This creates the westerlies.

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And they're called the
westerlies because they pull air

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from the West to the East

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and the bottom halves of
the polar convection cells

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take the cool air from the poles

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and sweep it to the 60
degree latitude lines

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and this creates the Easterlies winds.

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It's important to remember
that everything in this diagram

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is just an overall model.

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Global wind patterns are
even more complicated

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because water covered areas
and land covered areas

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absorb solar energy differently.

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These prevailing wind
patterns distribute heat

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and precipitation unevenly
between the tropics, temperate

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and polar regions of the earth.

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And this uneven distribution
creates different biomes.

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And this helps determine what
species can survive where.

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The tropical rainforest will
be in the low pressure areas

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near the equator.

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And right here, between
the polar and ferrel cells

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is another area of low pressure.

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Just like near the equator,
hot moist air rises here

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causing more precipitation
in the surrounding areas.

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So along this longitudinal line,

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you'll find many coniferous forests

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that thrive because of
all that precipitation.

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And there's a high
pressure area right here

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where the cool dry air sinks down

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so there's not as much precipitation.

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The air is drier here.

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You'll find many deserts
along this line and this line.

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So even though convection cells

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and prevailing winds are invisible,

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the ways they shape the
environment are not.


El Niño and La Niña

WEBVTT
Kind: captions
Language: en

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- [Instructor] Every few years
you might hear about El Nino

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in the news.

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And this also might come
with powerful images

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of flooding and rainfall,
but it is not just a storm.

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It's actually a climate
pattern that takes place

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in the Pacific Ocean.

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And we'll get a little bit more

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into what that actually means.

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Now, fun fact about how El Nino is named

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is that a long time ago,

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South American fishermen
noticed one December

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that the Pacific Ocean was actually warmer

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than it normally is.

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And this brought about
an abundance of fish.

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So because they were grateful,
they named this event

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El Nino to correlate
with the commemoration

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of Christ in this part of the
world during Christmas time.

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So that's a little trivia you
can keep in your back pocket,

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but back to what we hear
about El Nino in the news,

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you might also notice
that it isn't talked about

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every single year and that's
because El Nino comes around

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every two to seven years on average.

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And scientists are still not sure exactly

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what triggers El Nino, but
they know what signs to look

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for once it is approaching
or once we're in that event.

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So even though it's true

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that El Nino can bring about
heavy rainfall and flooding,

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it can also cause severe drought.

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So it's really important to
note that different regions

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around the world experience
different effects of El Nino.

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And we'll see a few examples of those.

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So sometimes an El Nino year is actually,

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a little bit helpful and might bring

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about some much needed rain,

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but other times bigger and
more severe El Nino events

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can bring devastating weather
events all across the globe.

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So we can look at the different effects

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by looking at the biggest
El Nino on record,

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during the 1997 to 1998 season.

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And we can see how different
regions were affected.

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So for example, in California,

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we saw very destructive mudslides.

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In Ecuador, there was heavy rainfall.

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And in Indonesia there was actually

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extreme drought and fire.

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And overall, this El Nino
event was really large

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and very destructive, there are estimates

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of about $36 billion in
damage to infrastructure.

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So you might be asking yourself,

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how can the same climate
pattern caused such drastically

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different effects around the globe?

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So to understand this better,

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we have to look at what a quote unquote

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normal year looks like.

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So this is a map of the Pacific Ocean.

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And normally there are winds
that are pushing warm water

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towards the West.

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So towards Asia and the Pacific islands

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and warm water accumulates on this side,

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on the other side on the East side,

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near Central and South America,

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we have an accumulation of cool water.

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And this is also due to a
process called upwelling,

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where cold water from the bottom
of the ocean is pushed up.

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And during these conditions in general,

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you would see less rain on this side.

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Or you'll see more rain on this side.

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Okay, so now let's look
at what changes during

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an El Nino year.

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So during an El Nino year,

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you still have trade
winds pushing water West,

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but they're much weaker
and that's the key here,

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that your trade winds are weakened,

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which means less water is pushed

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towards Asia and the Pacific islands.

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So compared to a normal year,

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you actually have cooler water
that starts to accumulate

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on this side of the Pacific Ocean.

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Meanwhile, on the other side,

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since less water is being
pushed away from Central

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and South America, you get
more warm water accumulating.

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So now you can think back to
those South American fishermen

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and why they were experiencing warm water

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along their coasts during an El Nino year.

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So this map does a really great job

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of showing the temperature gradient,

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and how this warm water
is staying along the coast

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of Central and South America
during an El Nino year.

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Meanwhile, you have cold water
accumulating towards Asia.

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And this causes a lot of changes

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that affect weather patterns.

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So in the US the Pacific jet stream

00:05:03.480 --> 00:05:06.193
moves South of its neutral position.

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And because of that, we start to see

00:05:08.970 --> 00:05:13.360
that the Northern US and
Canada are actually warmer

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and drier than they usually are.

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The US Gulf Coast and Southwest regions

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of the US are wetter,

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and are there more
increased risk for flooding.

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And South Asia and the Pacific
islands experience warmer

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and drier conditions that lead to drought

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on the other side of the Pacific Ocean.

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So now you start to see
how one event El Nino,

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this one climate pattern
can cause both flooding

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on one side of the Pacific Ocean,

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while also causing
drought on the other side.

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And finally, we'll end with La Nina,

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because you might hear this come up too

00:05:56.190 --> 00:05:57.711
when talking about El Nino.

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So La Nina is essentially the opposite.

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So if you've got El Nino down,

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you can start to understand what happens

00:06:04.090 --> 00:06:06.680
during La Nina event.

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So in this situation,

00:06:08.030 --> 00:06:13.030
our trade winds are getting
stronger than our normal years.

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So they're pushing water West,

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but they're doing so even more

00:06:18.780 --> 00:06:20.910
than we normally would expect.

00:06:20.910 --> 00:06:24.520
So this warm water that accumulates

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towards Asia actually
accumulates more West

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than it would during our normal years.

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And on the other side, we
have even more upwelling

00:06:37.280 --> 00:06:40.610
causing this cold water to accumulate.

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So the next time you hear El
Nino or LA Nina in the news,

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you'll know that they're not just talking

00:06:46.860 --> 00:06:50.050
about a really big storm,
but they're actually talking

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about climate patterns
that can affect weather

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all across the globe.


Soil Texture Triangle

WEBVTT
Kind: captions
Language: en

00:00:01.670 --> 00:00:04.150
- [Narrator] Today, we're
going to talk about soil.

00:00:04.150 --> 00:00:05.450
And you've probably noticed

00:00:05.450 --> 00:00:07.830
that there are many
different kinds of soils.

00:00:07.830 --> 00:00:09.160
The soil near a beach

00:00:09.160 --> 00:00:10.730
looks and feels very different

00:00:10.730 --> 00:00:12.620
than the soil in a forest.

00:00:12.620 --> 00:00:14.190
And part of the reason for that difference

00:00:14.190 --> 00:00:17.620
is something called soil texture.

00:00:17.620 --> 00:00:19.470
Soil texture.

00:00:19.470 --> 00:00:21.290
So when soil is formed,

00:00:21.290 --> 00:00:24.140
different types of rock break down

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because of the wind and
the rain and the weather.

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And they become differently
sized particles.

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And the combination of these
differently sized particles

00:00:33.740 --> 00:00:35.810
creates soil texture.

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So these differently sized particles

00:00:37.420 --> 00:00:39.740
can be broken down into three groups.

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We have sand.

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We have silt.

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And we have clay.

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Sand is made up of the larger
and heavier particles of soil.

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Sand is around two millimeters.

00:00:55.430 --> 00:01:00.350
Two millimeters to 0.05 millimeters.

00:01:00.350 --> 00:01:03.720
Which is a 20th of a
millimeter in diameter.

00:01:03.720 --> 00:01:06.553
And if it feels very gritty to touch.

00:01:08.570 --> 00:01:12.310
Clay on the other hand is the
smallest particles of soil.

00:01:12.310 --> 00:01:16.680
It's around 0.002
millimeters and the smaller.

00:01:16.680 --> 00:01:18.640
That's a 500th of a millimeter.

00:01:18.640 --> 00:01:20.083
So really tiny.

00:01:20.930 --> 00:01:22.650
If you rubbed clay between your fingers,

00:01:22.650 --> 00:01:23.690
it would feel smooth,

00:01:23.690 --> 00:01:24.720
and you wouldn't be able to feel

00:01:24.720 --> 00:01:26.950
the individual particles of soil.

00:01:26.950 --> 00:01:29.870
And unlike sand, you can't
even see clay's particles

00:01:29.870 --> 00:01:30.883
with a naked eye.

00:01:31.840 --> 00:01:35.240
Most people are familiar with
sandy soils and clay soils,

00:01:35.240 --> 00:01:37.550
but silt is right in between.

00:01:37.550 --> 00:01:40.810
It's made of a particles between
the size of clay and sand.

00:01:40.810 --> 00:01:45.410
So, it's between 0.05 millimeters

00:01:45.410 --> 00:01:48.283
and 0.002 millimeters.

00:01:50.610 --> 00:01:53.240
So when you think of silt
imagine baking flour.

00:01:53.240 --> 00:01:55.180
It's powdery.

00:01:55.180 --> 00:01:56.013
Powdery.

00:01:58.280 --> 00:02:01.210
And it can be carried
easily by wind and water.

00:02:01.210 --> 00:02:02.840
But soil is more complicated

00:02:02.840 --> 00:02:05.160
than just these three soil types.

00:02:05.160 --> 00:02:07.810
If you grabbed a handful of
soil from outside your home,

00:02:07.810 --> 00:02:10.880
it would probably be a combination
of all three soil types,

00:02:10.880 --> 00:02:13.010
along with some organic material.

00:02:13.010 --> 00:02:14.870
A soil's unique texture,

00:02:14.870 --> 00:02:18.040
that is it's combination
of silt, sand, and clay,

00:02:18.040 --> 00:02:19.840
affects how plants will grow.

00:02:19.840 --> 00:02:22.560
And so gardeners often wanna
know about their soil textures.

00:02:22.560 --> 00:02:25.210
They know what kinds of
plants to grow in that area.

00:02:25.210 --> 00:02:26.600
This is where a helpful diagram

00:02:26.600 --> 00:02:29.610
called the soil texture triangle comes in.

00:02:29.610 --> 00:02:31.713
The soil texture triangle.

00:02:32.910 --> 00:02:35.660
The triangle allows us
to place any soil sample

00:02:35.660 --> 00:02:38.670
into one of 12 different
soil texture categories.

00:02:38.670 --> 00:02:40.850
The 12 different categories are broken up

00:02:40.850 --> 00:02:43.150
based on the percentage of silt, clay

00:02:43.150 --> 00:02:44.780
and sand in the soil.

00:02:44.780 --> 00:02:45.790
To see how this works,

00:02:45.790 --> 00:02:47.640
let's do an example.

00:02:47.640 --> 00:02:50.973
Let's say a gardener determined
that a soil was 30% sand,

00:02:52.270 --> 00:02:53.513
40% silt,

00:02:54.430 --> 00:02:56.253
and 30% clay.

00:02:57.310 --> 00:03:00.080
So, how can we figure out
what kind of soil have?

00:03:00.080 --> 00:03:03.570
First, we need to pick a side
of the triangle to start with.

00:03:03.570 --> 00:03:05.070
I'll start with sand.

00:03:05.070 --> 00:03:07.160
We know we have 30% sand.

00:03:07.160 --> 00:03:09.800
So, we go along the side of the triangle,

00:03:09.800 --> 00:03:12.620
until we find that 30% mark, right here.

00:03:12.620 --> 00:03:14.630
We want to draw a line from this point,

00:03:14.630 --> 00:03:15.540
to the side of the triangle,

00:03:15.540 --> 00:03:17.790
that this arrow is pointing to.

00:03:17.790 --> 00:03:20.070
The arrow is pointing
towards the clay side.

00:03:20.070 --> 00:03:23.570
So we draw the line
from the 30% sand mark,

00:03:23.570 --> 00:03:26.120
through the triangle, to the clay side.

00:03:26.120 --> 00:03:26.973
Like this.

00:03:27.850 --> 00:03:29.080
And by drawing this line,

00:03:29.080 --> 00:03:31.190
we already know that our soil sample

00:03:31.190 --> 00:03:34.030
will fall into a category along this line.

00:03:34.030 --> 00:03:39.000
So, it could be so silt loam,
loam, clay loam, or clay.

00:03:39.000 --> 00:03:41.020
But to figure out exactly where it is,

00:03:41.020 --> 00:03:42.810
let's draw another line.

00:03:42.810 --> 00:03:44.220
Let's do silt.

00:03:44.220 --> 00:03:46.153
The soil has 40% silt.

00:03:47.030 --> 00:03:48.900
We'll draw our line through the triangle

00:03:48.900 --> 00:03:51.263
towards the sand side
because of this arrow.

00:03:52.690 --> 00:03:54.750
So again, we do the same thing.

00:03:54.750 --> 00:03:58.030
We go along the percent silt's line.

00:03:58.030 --> 00:04:00.160
We find the 40% mark.

00:04:00.160 --> 00:04:03.270
And we're gonna draw a line from 40%

00:04:03.270 --> 00:04:05.063
to the other side of the triangle.

00:04:06.020 --> 00:04:07.920
And you can see when we draw that line,

00:04:07.920 --> 00:04:11.323
these two lines intersect
at the clay loam point.

00:04:12.300 --> 00:04:13.150
But to double-check,

00:04:13.150 --> 00:04:18.150
let's draw a line from the 30% clay mark,

00:04:19.300 --> 00:04:23.590
to the other side of
the triangle, like this.

00:04:23.590 --> 00:04:25.470
As you can see, these three lines,

00:04:25.470 --> 00:04:27.300
all intersected at the same point,

00:04:27.300 --> 00:04:28.603
right here in clay loam.

00:04:30.020 --> 00:04:31.943
So we know that we have clay loam.

00:04:32.970 --> 00:04:36.250
Notice that the percentage
of clay, silt and sand,

00:04:36.250 --> 00:04:38.860
all add up to a total of 100%.

00:04:38.860 --> 00:04:40.960
We could have picked
any two of these lines,

00:04:40.960 --> 00:04:43.090
just two, to find our answer.

00:04:43.090 --> 00:04:45.810
We don't need to draw
three lines every time.

00:04:45.810 --> 00:04:47.610
Let's do another example.

00:04:47.610 --> 00:04:50.370
Let's say our gardener has another plot,

00:04:50.370 --> 00:04:53.593
with soil that contains 58% sand,

00:04:54.560 --> 00:04:55.873
27% silt,

00:04:57.770 --> 00:04:59.693
and 15% clay.

00:05:00.890 --> 00:05:05.060
So, I'm gonna go along the
percent sand part and find 58%.

00:05:05.060 --> 00:05:07.400
So that's pretty close to 60.

00:05:07.400 --> 00:05:09.490
And I'm gonna draw a line from 58%,

00:05:09.490 --> 00:05:11.740
to the other side of the triangle.

00:05:11.740 --> 00:05:13.260
Sometimes the example question,

00:05:13.260 --> 00:05:15.630
doesn't give us round
numbers to work with.

00:05:15.630 --> 00:05:16.520
So when that's the case,

00:05:16.520 --> 00:05:18.270
I like to use a ruler or a straight edge,

00:05:18.270 --> 00:05:21.000
just to make sure that my
lines are in the correct spot.

00:05:21.000 --> 00:05:24.500
Next I'm gonna use the 15% clay.

00:05:24.500 --> 00:05:27.160
So I find the 15% clay mark,

00:05:27.160 --> 00:05:30.810
and we draw a line for 15% to
the other side of the triangle

00:05:30.810 --> 00:05:33.750
following the direction of
the lines within the triangle.

00:05:33.750 --> 00:05:36.660
And we can see that our two lines

00:05:36.660 --> 00:05:39.470
intersect at this point.

00:05:39.470 --> 00:05:41.690
So, we have sandy loam.

00:05:41.690 --> 00:05:46.690
The gardener has these two
plots clay loam and sandy loam.

00:05:48.550 --> 00:05:50.540
So when the gardener
looks at her two plots,

00:05:50.540 --> 00:05:52.160
she might wanna take into account

00:05:52.160 --> 00:05:53.670
the different kinds of soil.

00:05:53.670 --> 00:05:56.230
Certain kinds of plants do
better in clay loam soil

00:05:56.230 --> 00:05:58.970
while others would do better
in the sandy loam soil.

00:05:58.970 --> 00:06:00.960
This is partly because clay soils,

00:06:00.960 --> 00:06:04.050
hold onto moisture for
longer than sandy soils do.

00:06:04.050 --> 00:06:06.520
So the plants in the
gardeners sandy loam plot,

00:06:06.520 --> 00:06:08.610
would do better if they
had longer root systems

00:06:08.610 --> 00:06:10.800
to access water deeper in the ground.

00:06:10.800 --> 00:06:14.330
Or if they stored water inside
themselves to access later.

00:06:14.330 --> 00:06:16.690
The gardener could use a
gardening book or the internet

00:06:16.690 --> 00:06:18.340
to see what kinds of plants would do best

00:06:18.340 --> 00:06:20.320
in each of her two plots.

00:06:20.320 --> 00:06:23.353
Thanks for watching and
happy soil identification.


Hess's law

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:03.040
- [Instructor] Hess's law
states that the overall change

00:00:03.040 --> 00:00:06.270
in enthalpy for a chemical
reaction is equal to the sum

00:00:06.270 --> 00:00:09.333
of the enthalpy changes for each step.

00:00:09.333 --> 00:00:12.020
And this is independent of the path taken.

00:00:12.020 --> 00:00:14.900
So it doesn't matter what
set of reactions you use.

00:00:14.900 --> 00:00:18.660
If you add up those reactions
and they equal the reaction

00:00:18.660 --> 00:00:19.891
that you're trying to find,

00:00:19.891 --> 00:00:22.270
you can also sum the enthalpies

00:00:22.270 --> 00:00:25.090
to find the enthalpy
change for the reaction.

00:00:25.090 --> 00:00:27.440
As an example, let's say we're
trying to find the change

00:00:27.440 --> 00:00:31.490
in enthalpy for the reaction
of carbon with hydrogen gas

00:00:31.490 --> 00:00:35.210
to form C2H2, which is acetylene.

00:00:35.210 --> 00:00:38.080
We can calculate the change
in enthalpy for the formation

00:00:38.080 --> 00:00:42.600
of acetylene using these
three reactions below.

00:00:42.600 --> 00:00:45.460
Our approach will involve
looking at these three reactions

00:00:45.460 --> 00:00:48.440
and comparing them to
the original reaction

00:00:48.440 --> 00:00:50.940
to see if we need to change anything.

00:00:50.940 --> 00:00:53.690
For example, if we look at reaction one,

00:00:53.690 --> 00:00:55.480
there's one mole of acetylene

00:00:55.480 --> 00:00:57.900
on the left side of the equation.

00:00:57.900 --> 00:01:00.980
And if we compare that
to the original reaction,

00:01:00.980 --> 00:01:03.670
there's one mole of
acetylene on the right side

00:01:03.670 --> 00:01:04.870
of the equation.

00:01:04.870 --> 00:01:07.480
So we need to reverse equation one

00:01:07.480 --> 00:01:11.240
to make it look more like
our original reaction.

00:01:11.240 --> 00:01:14.480
To save time, I've gone ahead
and reversed equation one.

00:01:14.480 --> 00:01:16.840
So you can see, I did that down here.

00:01:16.840 --> 00:01:19.820
Looking at the original
equation for equation one,

00:01:19.820 --> 00:01:21.160
here where the products

00:01:21.160 --> 00:01:24.730
and now we've made those
products the reactants.

00:01:24.730 --> 00:01:28.110
And what were the reactants
over here for equation one

00:01:28.110 --> 00:01:30.960
have now become the products.

00:01:30.960 --> 00:01:33.520
The change in enthalpy for equation one

00:01:33.520 --> 00:01:38.520
is -1,299.6 kilojoules
per mole of reaction.

00:01:38.740 --> 00:01:41.780
Kilojoules per mole reaction
just means how the reaction

00:01:41.780 --> 00:01:43.670
is written in the balanced equation.

00:01:43.670 --> 00:01:46.940
And since we reversed equation one,

00:01:46.940 --> 00:01:50.240
we also need to reverse
the sign for Delta H.

00:01:50.240 --> 00:01:52.800
So instead of this being a negative,

00:01:52.800 --> 00:01:53.840
instead of this being a negative,

00:01:53.840 --> 00:01:56.350
we're gonna go ahead and
change this into a positive.

00:01:56.350 --> 00:02:00.000
And also let's go ahead and
cross out the first equation.

00:02:00.000 --> 00:02:03.290
So we don't get confused.

00:02:03.290 --> 00:02:05.010
Next, we look at equation two

00:02:05.010 --> 00:02:07.360
and we compare it to our original.

00:02:07.360 --> 00:02:10.540
For equation two there's
one mole of solid carbon

00:02:10.540 --> 00:02:14.608
on the left side and looking
at our original reaction,

00:02:14.608 --> 00:02:18.200
there's two moles of
carbon on the left side.

00:02:18.200 --> 00:02:21.670
So to get equation two, to
look like our original equation

00:02:21.670 --> 00:02:25.710
we need to multiply everything
through, by a factor of two.

00:02:25.710 --> 00:02:29.080
So we're gonna multiply
everything in equation two

00:02:29.080 --> 00:02:30.623
by a factor of two.

00:02:31.630 --> 00:02:33.540
To save some time, I have
gone ahead and written out

00:02:33.540 --> 00:02:34.373
what we would get.

00:02:34.373 --> 00:02:37.330
We would get two carbons plus two O2s

00:02:37.330 --> 00:02:39.850
goes to 2CO2.

00:02:39.850 --> 00:02:42.170
The change in the
enthalpy for the formation

00:02:42.170 --> 00:02:44.350
of one mole of CO2

00:02:44.350 --> 00:02:49.050
was -393.5 kilojoules
per mole of reaction.

00:02:49.050 --> 00:02:52.250
But now we're forming two moles of CO2.

00:02:52.250 --> 00:02:55.070
And since we multiplied
the equation through

00:02:55.070 --> 00:02:56.860
by a factor of two, we also need

00:02:56.860 --> 00:02:58.910
to multiply the change in enthalpy

00:02:58.910 --> 00:03:01.200
by a factor of two as well.

00:03:01.200 --> 00:03:03.810
And also let's go ahead and cross out

00:03:03.810 --> 00:03:04.859
this first version here

00:03:04.859 --> 00:03:08.333
because now we're
forming two moles of CO2.

00:03:09.170 --> 00:03:10.670
Next, we look at equation three

00:03:10.670 --> 00:03:12.870
and we can see there's
one mole of hydrogen gas

00:03:12.870 --> 00:03:14.700
on the left side of the equation

00:03:14.700 --> 00:03:16.770
which matches the original reaction

00:03:16.770 --> 00:03:20.040
which also has one mole of
hydrogen gas on the left side.

00:03:20.040 --> 00:03:22.840
So we don't need to do
anything to equation three.

00:03:22.840 --> 00:03:25.020
And since we're not doing
anything to the equation,

00:03:25.020 --> 00:03:26.750
we're also not gonna do anything

00:03:26.750 --> 00:03:28.980
to the change in the enthalpy.

00:03:28.980 --> 00:03:33.980
So it's gonna stay -285.8
kilojoules per mole of reaction.

00:03:34.270 --> 00:03:37.210
Next we add up all of our
reactants and products.

00:03:37.210 --> 00:03:42.210
So we have two CO2 plus
H2O plus 2C plus 2O2

00:03:46.060 --> 00:03:48.830
plus H2 plus one half O2.

00:03:48.830 --> 00:03:52.460
So those are all written
down here for our reactants.

00:03:52.460 --> 00:03:56.210
And then for the products, let
me just change colors here.

00:03:56.210 --> 00:03:59.400
We have C2H2 plus 5O2

00:04:00.470 --> 00:04:04.410
plus 2CO2 plus H2O.

00:04:04.410 --> 00:04:08.623
And so those are written
over here for the products.

00:04:09.530 --> 00:04:11.310
Next we see what we can cancel out.

00:04:11.310 --> 00:04:13.220
There's 2CO2 on the left side

00:04:13.220 --> 00:04:15.250
and there's 2CO2 on the right side.

00:04:15.250 --> 00:04:16.540
So those cancel out.

00:04:16.540 --> 00:04:19.990
There's one water on the left
and one water on the right.

00:04:19.990 --> 00:04:23.598
And there's 2O2s plus one half O2

00:04:23.598 --> 00:04:28.430
which is 2.5O2s or five halves O2s.

00:04:28.430 --> 00:04:32.020
So the oxygen's cancel
out on both sides as well.

00:04:32.020 --> 00:04:37.020
And we can see we're left with
two carbons plus one hydrogen

00:04:37.390 --> 00:04:40.680
goes to form one C2H2

00:04:40.680 --> 00:04:45.560
which is the same as
our original equation.

00:04:45.560 --> 00:04:48.250
Since we were able to add up our equations

00:04:48.250 --> 00:04:50.310
and get the overall equation,

00:04:50.310 --> 00:04:51.640
according to Hess's law,

00:04:51.640 --> 00:04:55.310
we should also be able to
add the changes in enthalpies

00:04:55.310 --> 00:04:57.483
for these steps to get
the change in the enthalpy

00:04:57.483 --> 00:05:00.380
for the overall reaction.

00:05:00.380 --> 00:05:01.980
If we look at the changes in enthalpy

00:05:01.980 --> 00:05:06.910
for the individual steps, we had +1299.6

00:05:06.910 --> 00:05:08.090
for the first equation.

00:05:08.090 --> 00:05:09.470
And so that's up here.

00:05:09.470 --> 00:05:13.560
For the second equation we
had negative 393.5 times two,

00:05:13.560 --> 00:05:14.450
which is -787.

00:05:15.640 --> 00:05:19.570
And for our third equation, we had -285.8.

00:05:19.570 --> 00:05:22.120
So that's -285.8.

00:05:22.120 --> 00:05:23.360
When we add everything together

00:05:23.360 --> 00:05:28.360
we get +226.8 kilojoules
per mole of reaction.

00:05:29.600 --> 00:05:32.570
So for the formation of
one mole of acetylene

00:05:32.570 --> 00:05:35.920
from two moles of carbon
and one mole of hydrogen

00:05:35.920 --> 00:05:38.380
the change in enthalpy for this reaction

00:05:38.380 --> 00:05:43.163
is equal to +226.8 kilojoules
per mole of reaction.


Complex numbers with the same modulus (absolute value)

WEBVTT
Kind: captions
Language: en

00:00:00.200 --> 00:00:02.890
- [Instructor] We are asked,
which of these complex numbers

00:00:02.890 --> 00:00:06.520
has a modulus of 13?

00:00:06.520 --> 00:00:08.430
And just as a bit of a hint,

00:00:08.430 --> 00:00:12.520
when we're talking about the
modulus of a complex number,

00:00:12.520 --> 00:00:15.510
we're really just talking
about its absolute value.

00:00:15.510 --> 00:00:18.410
Or if we were to plot
it in the complex plane,

00:00:18.410 --> 00:00:20.100
which is what we have right over here,

00:00:20.100 --> 00:00:23.290
what is its distance from the origin?

00:00:23.290 --> 00:00:26.650
So really you need to find
which of these complex numbers

00:00:26.650 --> 00:00:31.650
has a distance of 13 from the
origin in the complex plane.

00:00:31.940 --> 00:00:34.540
Pause this video and see
if you can figure that out.

00:00:35.640 --> 00:00:38.240
All right, now let's work
through this together.

00:00:38.240 --> 00:00:40.640
Now one might jump out at you immediately

00:00:40.640 --> 00:00:43.880
that's going to have a
distance of 13 from the origin.

00:00:43.880 --> 00:00:46.350
If this is the origin right over here,

00:00:46.350 --> 00:00:49.180
we see that if we go exactly 13 units down

00:00:49.180 --> 00:00:52.500
we have this point right
over here, negative 13i.

00:00:52.500 --> 00:00:55.117
So immediately right
out of the gate, I say,

00:00:55.117 --> 00:00:58.270
"Okay, that complex number
has a modulus of 13,"

00:00:58.270 --> 00:01:00.450
but is that the only one?

00:01:00.450 --> 00:01:03.290
Well, we can actually visualize
all of the complex numbers

00:01:03.290 --> 00:01:06.770
that have a modulus of
13 by drawing a circle

00:01:06.770 --> 00:01:09.840
with the radius 13 centered at the origin.

00:01:09.840 --> 00:01:12.160
So let's do that.

00:01:12.160 --> 00:01:14.630
And we can see that it contains

00:01:14.630 --> 00:01:17.000
the first complex number
that we looked for,

00:01:17.000 --> 00:01:20.900
but it also seems to have included in it

00:01:20.900 --> 00:01:24.160
this one right over
here, and we can verify

00:01:24.160 --> 00:01:27.360
that the modulus right over
here is going to be 13.

00:01:27.360 --> 00:01:29.910
We can just use the Pythagorean theorem.

00:01:29.910 --> 00:01:33.730
So this distance right over here is 12.

00:01:33.730 --> 00:01:36.920
And this distance right over here is 5.

00:01:36.920 --> 00:01:38.840
And so we just need to figure out

00:01:38.840 --> 00:01:41.600
the hypotenuse right over here.

00:01:41.600 --> 00:01:43.910
And so we know that the hypotenuse

00:01:43.910 --> 00:01:47.660
is going to be the
square root of 5 squared

00:01:47.660 --> 00:01:49.240
plus 12 squared,

00:01:49.240 --> 00:01:53.637
which is equal to the
square root of 25 plus 144,

00:01:54.530 --> 00:01:57.670
which is equal to the square root of 169,

00:01:57.670 --> 00:02:01.050
which indeed does equal 13.

00:02:01.050 --> 00:02:03.160
So I like that choice as well.

00:02:03.160 --> 00:02:05.760
And we can see visually that
none of these other points

00:02:05.760 --> 00:02:08.600
that they already plotted
sit on that circle.

00:02:08.600 --> 00:02:10.710
So they don't have a modulus of 13.

00:02:10.710 --> 00:02:13.860
If we wanted to come up with
some other interesting points,

00:02:13.860 --> 00:02:17.640
we could instead of having
negative 5 plus 12i,

00:02:17.640 --> 00:02:20.270
we could have negative 5 minus 12i.

00:02:20.270 --> 00:02:21.770
It would get us right over there.

00:02:21.770 --> 00:02:24.480
And that would have a modulus of 13.

00:02:24.480 --> 00:02:27.310
And notice, when you have
your complex conjugate,

00:02:27.310 --> 00:02:29.260
it has the same modulus.

00:02:29.260 --> 00:02:30.760
Or you could go the other way around.

00:02:30.760 --> 00:02:35.510
Instead of negative 5 plus
12i, you could have 5 plus 12i.

00:02:35.510 --> 00:02:38.130
That also would have a modulates of 13.

00:02:38.130 --> 00:02:41.240
Or you could have 5 minus 12i.

00:02:41.240 --> 00:02:43.730
That also would have a modulus of 13.

00:02:43.730 --> 00:02:45.670
Now there's an infinite number of points,

00:02:45.670 --> 00:02:47.590
any of these points on the circle,

00:02:47.590 --> 00:02:50.323
that will have a modulus of 13.


Factoring polynomials using complex numbers

WEBVTT
Kind: captions
Language: en

00:00:00.340 --> 00:00:01.830
- [Instructor] We're told that Amat tried

00:00:01.830 --> 00:00:05.570
to write x to the fourth
plus 5x squared plus 4

00:00:05.570 --> 00:00:08.490
as a product of linear factors.

00:00:08.490 --> 00:00:10.140
This is his work.

00:00:10.140 --> 00:00:12.400
And then they'd tell us all
of the steps that he did.

00:00:12.400 --> 00:00:14.180
And then they say, in what step

00:00:14.180 --> 00:00:17.330
did Amat make his first mistake?

00:00:17.330 --> 00:00:20.203
So pause this video and see
if you can figure that out.

00:00:21.360 --> 00:00:24.190
All right, now let's work
through this together.

00:00:24.190 --> 00:00:26.530
So we're starting with x to
the fourth plus 10x squared

00:00:26.530 --> 00:00:27.760
plus 9.

00:00:27.760 --> 00:00:30.220
And it looks like Amat
tried to factor that

00:00:30.220 --> 00:00:33.570
into x squared plus 9
times x squared plus 1.

00:00:33.570 --> 00:00:35.920
And this indeed does make sense,

00:00:35.920 --> 00:00:38.100
because if we said that let's say,

00:00:38.100 --> 00:00:41.770
u is equal to x squared,

00:00:41.770 --> 00:00:43.840
we could rewrite this right over here

00:00:43.840 --> 00:00:48.097
as u squared plus 10u plus 9.

00:00:49.960 --> 00:00:51.640
The whole reason why you would do this is

00:00:51.640 --> 00:00:54.580
so that you could write
this higher order expression

00:00:54.580 --> 00:00:57.130
in terms of a second degree expression.

00:00:57.130 --> 00:00:59.090
And then we've learned
how to factor things like

00:00:59.090 --> 00:01:00.020
this many times.

00:01:00.020 --> 00:01:00.980
We look, we say, okay,

00:01:00.980 --> 00:01:03.570
what two numbers when I add them I get 10,

00:01:03.570 --> 00:01:05.620
and when I multiply them I get nine,

00:01:05.620 --> 00:01:07.270
and it would be nine and one?

00:01:07.270 --> 00:01:08.480
And so you could write this

00:01:08.480 --> 00:01:13.480
as u plus 9 times u plus 1.

00:01:13.800 --> 00:01:15.700
And of course, if u is equal to x squared,

00:01:15.700 --> 00:01:20.700
this would be x squared plus
9 times x squared plus 1.

00:01:21.220 --> 00:01:23.730
Which is exactly what
Amat has right over here.

00:01:23.730 --> 00:01:27.120
So step one is looking great.

00:01:27.120 --> 00:01:30.670
All right, now let's think
about what Amat did in step two.

00:01:30.670 --> 00:01:33.400
They didn't do anything
to x squared plus 9

00:01:33.400 --> 00:01:34.380
but it looks like they tried

00:01:34.380 --> 00:01:37.490
to further factor x squared plus 1.

00:01:37.490 --> 00:01:39.240
And this does seem right.

00:01:39.240 --> 00:01:41.560
We just have to remind ourselves just

00:01:41.560 --> 00:01:43.610
as you have a difference of squares

00:01:43.610 --> 00:01:45.620
if you're dealing with
non-complex numbers,

00:01:45.620 --> 00:01:47.540
so we could rewrite this right over here

00:01:47.540 --> 00:01:52.200
as x plus a times x minus a.

00:01:52.200 --> 00:01:54.840
We could have a sum of squares

00:01:54.840 --> 00:01:57.200
if we're thinking about complex numbers.

00:01:57.200 --> 00:02:02.200
This is going to be x
plus ai times x minus ai.

00:02:04.760 --> 00:02:08.170
And in this situation while the x is x

00:02:08.170 --> 00:02:10.880
and then our a would be one.

00:02:10.880 --> 00:02:13.760
So we're going to have x plus i,

00:02:13.760 --> 00:02:17.960
x plus i times x minus i .

00:02:17.960 --> 00:02:20.770
So step two is looking great.

00:02:20.770 --> 00:02:22.640
And now let's go do step three.

00:02:22.640 --> 00:02:24.150
So in step three,

00:02:24.150 --> 00:02:28.120
no change to this part of the expression.

00:02:28.120 --> 00:02:29.900
And it looks like Amat is trying

00:02:29.900 --> 00:02:33.280
to factor x squared plus 9
based on the same principle.

00:02:33.280 --> 00:02:35.480
Now x squared plus 9 is the same thing

00:02:35.480 --> 00:02:38.690
as x squared plus 3 squared.

00:02:38.690 --> 00:02:41.100
So if you use this exact same idea here,

00:02:41.100 --> 00:02:46.100
if you factored it should be
x plus 3i times x minus 3i.

00:02:48.420 --> 00:02:49.920
But what we see over here

00:02:49.920 --> 00:02:52.340
is Amat took the square root of three,

00:02:52.340 --> 00:02:54.090
instead of just having a three here.

00:02:54.090 --> 00:02:56.450
Amat treated it instead
of having a nine here

00:02:56.450 --> 00:02:58.790
as if we actually had a three

00:02:58.790 --> 00:03:01.730
so they made a little
bit of an error there.

00:03:01.730 --> 00:03:06.730
So this is the step where
Amat makes his first mistake

00:03:07.800 --> 00:03:08.633
and we're done.


Expected payoff example: protection plan

WEBVTT
Kind: captions
Language: en

00:00:00.160 --> 00:00:01.790
- [Instructor] We're told
that an electronic store

00:00:01.790 --> 00:00:05.000
gives customers the option of
purchasing a protection plan

00:00:05.000 --> 00:00:06.720
when customers buy a new television.

00:00:06.720 --> 00:00:08.330
That's actually quite common.

00:00:08.330 --> 00:00:10.990
The customer pays $80 for the plan and,

00:00:10.990 --> 00:00:14.180
if their television is
damaged or stops working,

00:00:14.180 --> 00:00:17.420
the store will replace it
for no additional charge.

00:00:17.420 --> 00:00:21.010
The store knows that 2% of
customers who buy this plan

00:00:21.010 --> 00:00:22.426
end up needing a replacement

00:00:22.426 --> 00:00:25.860
that costs the store $1,200 each.

00:00:25.860 --> 00:00:28.685
Here's a table that summarizes
the possible outcomes

00:00:28.685 --> 00:00:31.269
from the store's perspective.

00:00:31.269 --> 00:00:34.490
Let X represent the store's net gain

00:00:34.490 --> 00:00:36.580
from one of these plans.

00:00:36.580 --> 00:00:39.680
Calculate the expected net gain.

00:00:39.680 --> 00:00:41.850
So pause this video, see if
you can have a go at that

00:00:41.850 --> 00:00:43.700
before we work through this together.

00:00:44.860 --> 00:00:46.780
So we have the two scenarios here.

00:00:46.780 --> 00:00:49.060
The first scenario is
that the store does need

00:00:49.060 --> 00:00:51.230
to replace the TV
because something happens

00:00:51.230 --> 00:00:53.660
and so it's gonna cost
$1,200 to the store.

00:00:53.660 --> 00:00:56.300
But remember they got $80
for the protection plan.

00:00:56.300 --> 00:00:59.900
So you have a net gain of negative $1,120

00:00:59.900 --> 00:01:01.520
from the store's perspective.

00:01:01.520 --> 00:01:02.540
There's the other scenario,

00:01:02.540 --> 00:01:03.960
which is more favorable for the store,

00:01:03.960 --> 00:01:07.330
which is a customer does
not need a replacement TV,

00:01:07.330 --> 00:01:09.360
so that has no cost and so their net gain

00:01:09.360 --> 00:01:11.520
is just the $80 for the plan.

00:01:11.520 --> 00:01:13.370
So to figure out the expected net gain,

00:01:13.370 --> 00:01:14.750
we just have to figure
out the probabilities

00:01:14.750 --> 00:01:17.400
of each of these and take
the weighted average of them.

00:01:17.400 --> 00:01:19.050
So what's the probability
that they will have

00:01:19.050 --> 00:01:20.640
to replace the TV?

00:01:20.640 --> 00:01:22.930
Well, we know 2% of
customers who buy this plan

00:01:22.930 --> 00:01:24.640
end up needing a replacement.

00:01:24.640 --> 00:01:26.840
So we could say this is two over 100

00:01:26.840 --> 00:01:29.300
or maybe I'll write it as 0.02.

00:01:29.300 --> 00:01:32.170
This is the probability of X.

00:01:32.170 --> 00:01:36.570
And then the probability of not
needing a replacement, 0.98.

00:01:36.570 --> 00:01:40.040
And so our expected net gain

00:01:40.040 --> 00:01:43.110
is going to be equal to the probability

00:01:43.110 --> 00:01:46.840
of needing replacement times
the net gain of a replacement.

00:01:46.840 --> 00:01:50.687
So it's going to be times -$1,120.

00:01:53.720 --> 00:01:56.150
And then we're gonna
have plus the probability

00:01:56.150 --> 00:01:58.980
of not needing replacement, which is 0.98

00:02:00.380 --> 00:02:01.870
times the net gain there.

00:02:01.870 --> 00:02:04.333
So that is $80.

00:02:05.690 --> 00:02:10.140
So we have 0.02x-1,120 is equal to that.

00:02:13.890 --> 00:02:15.850
And to that, we're gonna add,

00:02:15.850 --> 00:02:19.397
I'll open parentheses,
0.98x80, close parentheses,

00:02:23.170 --> 00:02:27.980
is going to be equal to $56.

00:02:27.980 --> 00:02:30.910
So this is equal to $56.

00:02:30.910 --> 00:02:33.410
And now you understand why the stores

00:02:33.410 --> 00:02:35.533
like to sell these replacement plans.


Expected payoff example: lottery ticket

WEBVTT
Kind: captions
Language: en

00:00:00.200 --> 00:00:01.870
- [Instructor] We're told
a Pick 4 lottery game

00:00:01.870 --> 00:00:05.120
involves drawing four numbered
balls from separate bins,

00:00:05.120 --> 00:00:08.530
each containing balls labeled from 0 to 9.

00:00:08.530 --> 00:00:11.500
So there are 10,000 possible
selections in total.

00:00:11.500 --> 00:00:12.333
For example, you could get

00:00:12.333 --> 00:00:14.530
a 0, a 0, a 0 and a 0,

00:00:14.530 --> 00:00:16.670
a 0, a 0, a 0 and a 1,

00:00:16.670 --> 00:00:21.080
all the way up to 9,999, four nines.

00:00:21.080 --> 00:00:23.890
Players can choose to play a
straight bet, where the player

00:00:23.890 --> 00:00:27.470
wins if they match all four
digits in the correct order.

00:00:27.470 --> 00:00:32.470
The lottery pays $4,500 on a
successful $1 straight bet.

00:00:33.520 --> 00:00:38.520
Let X represent a player's
net gain on a $1 straight bet.

00:00:39.580 --> 00:00:42.860
Calculate the expected net gain.

00:00:42.860 --> 00:00:46.310
And they say, hint, the expected
net gain can be negative.

00:00:46.310 --> 00:00:47.610
So why don't you pause this video

00:00:47.610 --> 00:00:50.363
and see if you can calculate
the expected net gain?

00:00:51.780 --> 00:00:52.613
All right.

00:00:52.613 --> 00:00:54.570
So there's a couple of ways
that we can approach this.

00:00:54.570 --> 00:00:57.940
One way is to just think about
the two different outcomes.

00:00:57.940 --> 00:01:01.430
There's a scenario where you
win with your straight bet.

00:01:01.430 --> 00:01:04.420
There's a scenario where you
lose with your straight bet.

00:01:04.420 --> 00:01:05.253
Now let's think about

00:01:05.253 --> 00:01:09.090
the net gain in either
one of those scenarios.

00:01:09.090 --> 00:01:12.420
The scenario where you win, you pay $1,

00:01:12.420 --> 00:01:16.950
we know it's a $1 straight
bet, and you get $4,500.

00:01:16.950 --> 00:01:18.750
So what's the net gain?

00:01:18.750 --> 00:01:22.160
So it's going to be $4,500 minus one.

00:01:22.160 --> 00:01:24.887
So your net gain is going to be $4,499.

00:01:29.040 --> 00:01:32.020
Now what about the net gain in
the situation that you lose?

00:01:32.020 --> 00:01:33.880
Well, in the situation that you lose,

00:01:33.880 --> 00:01:35.280
you just lose a dollar.

00:01:35.280 --> 00:01:40.280
So this is just going to be
negative $1 right over here.

00:01:40.860 --> 00:01:41.693
Now let's think about

00:01:41.693 --> 00:01:44.230
the probabilities of
each of these situations.

00:01:44.230 --> 00:01:47.550
So the probability, so
the probability of a win

00:01:47.550 --> 00:01:49.770
we know is 1 in 10,000,

00:01:49.770 --> 00:01:52.800
1 in 10,000.

00:01:52.800 --> 00:01:55.010
And what's the probability of a loss?

00:01:55.010 --> 00:01:58.540
Well, that's going to be 9,999

00:01:58.540 --> 00:02:00.990
out of 10,000.

00:02:00.990 --> 00:02:02.984
And then our expected net gain is just

00:02:02.984 --> 00:02:05.320
going to be the weighted
average of these two.

00:02:05.320 --> 00:02:08.960
So I could write our expected net gain

00:02:08.960 --> 00:02:12.940
is going to be 4,499

00:02:12.940 --> 00:02:15.780
times the probability of that, 1 in 10,000

00:02:16.810 --> 00:02:19.030
plus negative 1 times this,

00:02:19.030 --> 00:02:20.020
so that I could just write that

00:02:20.020 --> 00:02:25.020
as minus 9,999 over 10,000.

00:02:26.250 --> 00:02:28.730
And so this is going to
be equal to, let's see,

00:02:28.730 --> 00:02:30.470
it's going to be 4,499

00:02:31.730 --> 00:02:35.120
minus 9,999,

00:02:35.120 --> 00:02:38.970
all of that over 10,000.

00:02:38.970 --> 00:02:43.380
And let's see, this is
going to be equal to

00:02:43.380 --> 00:02:45.943
negative 5,500 over 10,000,

00:02:47.028 --> 00:02:50.607
negative 5,500 over 10,000,

00:02:51.570 --> 00:02:55.410
which is the same thing
as negative 55 over 100,

00:02:55.410 --> 00:02:56.880
or I could write it this way.

00:02:56.880 --> 00:03:00.150
This is equal to negative .55.

00:03:00.150 --> 00:03:03.780
I could write it this way, 0.55.

00:03:03.780 --> 00:03:07.010
So that's one way to calculate
the expected net gain.

00:03:07.010 --> 00:03:09.070
Another way to approach
it is to say, all right,

00:03:09.070 --> 00:03:11.280
what if we were to get 10,000 tickets?

00:03:11.280 --> 00:03:15.280
What is our expected net
gain on the 10,000 tickets?

00:03:15.280 --> 00:03:19.460
Well, we would pay $10,000

00:03:21.120 --> 00:03:22.970
and we would expect to win once.

00:03:22.970 --> 00:03:25.830
It's not a guarantee, but
we would expect to win once.

00:03:25.830 --> 00:03:27.150
So expect

00:03:29.250 --> 00:03:32.570
4,500 in payout.

00:03:32.570 --> 00:03:36.600
And so you would then, let's
see, you would have a net gain

00:03:37.840 --> 00:03:41.860
of, it would be negative $5,500,

00:03:41.860 --> 00:03:45.240
negative $5,500.

00:03:45.240 --> 00:03:48.910
Now this is the net gain
when you do 10,000 tickets.

00:03:48.910 --> 00:03:52.130
Now, if you wanted to find the
expected net gain per ticket

00:03:52.130 --> 00:03:53.890
you would then just divide by 10,000.

00:03:53.890 --> 00:03:54.780
And if you did that,

00:03:54.780 --> 00:03:58.760
you would get exactly what we
just calculated the other way.

00:03:58.760 --> 00:04:00.950
So any way you try to approach this

00:04:00.950 --> 00:04:02.693
this is not a great bet.


Interpreting expected value

WEBVTT
Kind: captions
Language: en

00:00:00.400 --> 00:00:03.380
- [Instructor] We're told a
certain lottery ticket costs $2

00:00:03.380 --> 00:00:05.057
and the back of the ticket says,

00:00:05.057 --> 00:00:08.020
"The overall odds of winning
a prize with this ticket

00:00:08.020 --> 00:00:09.500
are one to 50,

00:00:09.500 --> 00:00:13.530
and the expected return
for this ticket is $0.95."

00:00:13.530 --> 00:00:16.650
Which interpretations of the
expected value are correct?

00:00:16.650 --> 00:00:19.300
Choose all answers that apply.

00:00:19.300 --> 00:00:21.570
Pause this video, have a go at that.

00:00:21.570 --> 00:00:24.370
All right, now let's go
through each of these choices.

00:00:24.370 --> 00:00:26.160
So choice A says the probability

00:00:26.160 --> 00:00:31.160
that one of these tickets wins
a prize is 0.95 on average.

00:00:32.870 --> 00:00:34.940
Well, I see where they're
getting that 0.95.

00:00:34.940 --> 00:00:36.490
They're getting it from right over here,

00:00:36.490 --> 00:00:38.460
but that's not the probability
that you're winning,

00:00:38.460 --> 00:00:40.400
that's the expected return.

00:00:40.400 --> 00:00:42.830
The probability that
you win is much lower.

00:00:42.830 --> 00:00:44.750
If the odds are one to 50,

00:00:44.750 --> 00:00:49.340
that means that the probability
of winning is one to 51.

00:00:49.340 --> 00:00:52.490
So it's a much lower probability
than this right over here.

00:00:52.490 --> 00:00:54.530
So definitely rule that out.

00:00:54.530 --> 00:00:58.443
Someone who buys this ticket
is most likely to win $0.95.

00:01:00.390 --> 00:01:03.890
That is not necessarily the case either.

00:01:03.890 --> 00:01:08.660
We don't know what the different
outcomes are for the prize.

00:01:08.660 --> 00:01:10.760
It's very likely that there's no outcome

00:01:10.760 --> 00:01:13.420
for that prize where
you win exactly $0.95.

00:01:13.420 --> 00:01:16.610
Instead, there's likely to be
outcomes that are much larger

00:01:16.610 --> 00:01:18.550
than that with very low probabilities,

00:01:18.550 --> 00:01:20.440
and then when you take
the weighted average

00:01:20.440 --> 00:01:22.310
of all of the outcomes,

00:01:22.310 --> 00:01:25.690
then you get an expected return of $0.95.

00:01:25.690 --> 00:01:29.490
So it's actually maybe even
impossible to win exactly $0.95.

00:01:29.490 --> 00:01:31.410
So I would rule that out.

00:01:31.410 --> 00:01:33.080
If we looked at many of these tickets,

00:01:33.080 --> 00:01:37.523
the average return would
be about $0.95 per ticket.

00:01:38.370 --> 00:01:40.260
That one feels pretty interesting,

00:01:40.260 --> 00:01:42.460
'cause we're looking at
many of these tickets.

00:01:42.460 --> 00:01:46.720
And so across many of them, you
would expect to, on average,

00:01:46.720 --> 00:01:49.430
get the expected return as your return.

00:01:49.430 --> 00:01:51.720
And so this is what we are seeing here.

00:01:51.720 --> 00:01:53.370
The average return would be about that.

00:01:53.370 --> 00:01:54.600
It would be approximately that.

00:01:54.600 --> 00:01:56.140
So I like that choice.

00:01:56.140 --> 00:01:58.980
That is a good interpretation
of expected value.

00:01:58.980 --> 00:02:00.070
And then choice D,

00:02:00.070 --> 00:02:03.000
if 1,000 people each bought
one of these tickets,

00:02:03.000 --> 00:02:08.000
they'd expect a net gain
of about $950 in total.

00:02:09.100 --> 00:02:10.810
This one is tempting.

00:02:10.810 --> 00:02:14.540
Instead of net gain,
if it just said return,

00:02:14.540 --> 00:02:15.710
this would make a lot of sense.

00:02:15.710 --> 00:02:18.540
In fact, it would be completely
consistent with choice C.

00:02:18.540 --> 00:02:21.250
If you have 1,000 people,
that would be many tickets,

00:02:21.250 --> 00:02:23.500
and if on average, if their average return

00:02:23.500 --> 00:02:25.560
is about $0.95 per ticket,

00:02:25.560 --> 00:02:29.210
then their total return
would be about $950,

00:02:29.210 --> 00:02:32.550
but they didn't write return
here, they wrote net gain.

00:02:32.550 --> 00:02:36.640
Net gain would be how much you
get minus how much you paid.

00:02:36.640 --> 00:02:38.320
And 1,000 people would have to pay,

00:02:38.320 --> 00:02:41.530
if they each got a
ticket, would pay $2,000.

00:02:41.530 --> 00:02:43.210
So they would pay 2,000.

00:02:43.210 --> 00:02:46.590
They would expect a return of $950.

00:02:46.590 --> 00:02:50.690
Their net gain would
actually be negative $1,050.

00:02:50.690 --> 00:02:53.113
So we would rule that one out as well.


Probability distributions from empirical data

WEBVTT
Kind: captions
Language: en

00:00:00.270 --> 00:00:02.140
- [Instructor] We're told
that Jada owns a restaurant

00:00:02.140 --> 00:00:05.200
where customers can make
their orders using an app.

00:00:05.200 --> 00:00:07.130
She decides to offer a discount

00:00:07.130 --> 00:00:09.610
on appetizers to attract more customers.

00:00:09.610 --> 00:00:12.130
And she's curious about the probability

00:00:12.130 --> 00:00:16.690
that a customer orders a
large number of appetizers.

00:00:16.690 --> 00:00:19.070
Jada tracked how many appetizers

00:00:19.070 --> 00:00:22.950
were in each of the past 500 orders.

00:00:22.950 --> 00:00:24.810
All right, so the number of appetizers,

00:00:24.810 --> 00:00:27.760
so 40 out of the 500
ordered zero appetizers,

00:00:27.760 --> 00:00:32.030
and for example, 120 out of the
500 ordered three appetizers

00:00:32.030 --> 00:00:33.600
and so on and so forth.

00:00:33.600 --> 00:00:37.990
Let X represent the number of
appetizers in a random order.

00:00:37.990 --> 00:00:40.560
Based on these results,
construct an approximate

00:00:40.560 --> 00:00:43.230
probability distribution of X.

00:00:43.230 --> 00:00:45.530
Pause this video and see if
you can have a go at this

00:00:45.530 --> 00:00:47.093
before we do this together.

00:00:48.030 --> 00:00:51.640
All right, so they're
telling us an approximate

00:00:51.640 --> 00:00:54.700
probability distribution,
because we don't know

00:00:54.700 --> 00:00:56.560
the actual probability.

00:00:56.560 --> 00:00:58.180
We can't get into people's minds

00:00:58.180 --> 00:01:00.530
and figure out the probability
that the neurons fire

00:01:00.530 --> 00:01:03.840
in exactly the right
way to order appetizers.

00:01:03.840 --> 00:01:06.880
But what we can do is
look at past results,

00:01:06.880 --> 00:01:09.260
empirical data right over here

00:01:09.260 --> 00:01:11.860
to approximate the distribution.

00:01:11.860 --> 00:01:14.810
So what we can do is look at the last 500,

00:01:14.810 --> 00:01:16.390
and for each of the outcomes

00:01:16.390 --> 00:01:19.710
think about what fraction of
the last 500 had that outcome.

00:01:19.710 --> 00:01:21.910
And that will be our approximation.

00:01:21.910 --> 00:01:24.610
And so the outcomes here, we
could have zero appetizers,

00:01:24.610 --> 00:01:29.610
one, two, three, four, five, or six.

00:01:29.680 --> 00:01:32.356
Now the approximate probability

00:01:32.356 --> 00:01:36.280
of zero appetizers is
going to be 40 over 500,

00:01:36.280 --> 00:01:40.140
which is the same thing as four over 50,

00:01:40.140 --> 00:01:44.160
which is the same thing as two over 25.

00:01:44.160 --> 00:01:47.410
So I'll write two 25th right over there.

00:01:47.410 --> 00:01:49.997
The probability of one appetizer,

00:01:49.997 --> 00:01:52.090
well, that's going to be 90, the over 500,

00:01:52.090 --> 00:01:55.390
which is the same thing as nine over 50.

00:01:55.390 --> 00:01:58.450
And I think that's
already in lowest terms.

00:01:58.450 --> 00:02:03.450
Then 160 over 500 is the
same thing as 16 over 50,

00:02:05.580 --> 00:02:08.773
which is the same thing as eight over 25.

00:02:10.546 --> 00:02:11.700
And we just keep going.

00:02:11.700 --> 00:02:16.700
120 out of 500 is the same
thing as 12 out of 50,

00:02:17.000 --> 00:02:19.263
or six out of 25.

00:02:20.444 --> 00:02:25.444
Six out of 25, and then 50 out of 500.

00:02:25.750 --> 00:02:27.510
Well, that's one out of every 10.

00:02:27.510 --> 00:02:29.900
So I'll just write it like that.

00:02:29.900 --> 00:02:34.900
30 out of 500 is the same
thing as three out of 50.

00:02:36.120 --> 00:02:38.370
I'll just write it like that.

00:02:38.370 --> 00:02:40.950
And that last but not least, 10 out of 500

00:02:40.950 --> 00:02:43.303
is the same thing as one in 50.

00:02:44.150 --> 00:02:45.330
And we're done.

00:02:45.330 --> 00:02:47.420
We have just constructed an approximate

00:02:47.420 --> 00:02:50.853
probability distribution
for our random variable X.


Theoretical probability distribution example: multiplication

WEBVTT
Kind: captions
Language: en

00:00:00.290 --> 00:00:02.180
- [Instructor] We're told
that Kai goes to a restaurant

00:00:02.180 --> 00:00:04.417
that advertises a promotion saying,

00:00:04.417 --> 00:00:07.910
"1 in 5 customers get a free dessert!"

00:00:07.910 --> 00:00:12.050
Suppose Kai goes to the
restaurant twice in a given week,

00:00:12.050 --> 00:00:15.330
and each time he has a 1/5 probability

00:00:15.330 --> 00:00:17.520
of getting a free dessert.

00:00:17.520 --> 00:00:21.400
Let X represent the
number of free desserts

00:00:21.400 --> 00:00:23.980
Kai gets in his two trips.

00:00:23.980 --> 00:00:28.560
Construct the theoretical
probability distribution of X.

00:00:28.560 --> 00:00:30.220
All right, so pause this video

00:00:30.220 --> 00:00:31.670
and see if you can work through this

00:00:31.670 --> 00:00:33.333
before we do it together.

00:00:34.260 --> 00:00:36.380
All right, so first let's just think about

00:00:36.380 --> 00:00:38.930
the possible values that X could take on.

00:00:38.930 --> 00:00:40.950
This is the number of
free desserts he gets,

00:00:40.950 --> 00:00:42.630
and he visits twice.

00:00:42.630 --> 00:00:44.760
So there's some world in which

00:00:44.760 --> 00:00:46.610
he doesn't get any free desserts.

00:00:46.610 --> 00:00:50.090
So that's 0 in his two visits.

00:00:50.090 --> 00:00:51.340
Maybe on one of the visits,

00:00:51.340 --> 00:00:53.410
he gets a dessert and
the other one he doesn't,

00:00:53.410 --> 00:00:55.170
and maybe in both of his visits

00:00:55.170 --> 00:00:57.420
he actually is able to get a free dessert.

00:00:57.420 --> 00:00:58.810
So he's going to have someplace

00:00:58.810 --> 00:01:02.310
from 0 to 2 free desserts in a given week.

00:01:02.310 --> 00:01:03.143
So we just have to figure out

00:01:03.143 --> 00:01:05.420
the probability of each of these.

00:01:05.420 --> 00:01:08.630
So let's first of all think
about the probability,

00:01:08.630 --> 00:01:10.010
let me write it over here,

00:01:10.010 --> 00:01:13.710
the probability that
capital X is equal to 0

00:01:13.710 --> 00:01:15.600
is going to be equal to what?

00:01:15.600 --> 00:01:16.960
Well, that's going to be the probability

00:01:16.960 --> 00:01:20.200
that he doesn't get a
dessert on both days.

00:01:20.200 --> 00:01:21.460
And it's important to realize

00:01:21.460 --> 00:01:23.110
that these are independent events.

00:01:23.110 --> 00:01:24.387
It's not like the restaurant's gonna say,

00:01:24.387 --> 00:01:26.080
"Oh, if you didn't get
a dessert on one day,

00:01:26.080 --> 00:01:27.530
you're more likely to get
another the other day,"

00:01:27.530 --> 00:01:29.850
or, "Somehow if you got
it on a previous day,

00:01:29.850 --> 00:01:31.700
you're less likely than another day."

00:01:31.700 --> 00:01:33.610
That they are independent events.

00:01:33.610 --> 00:01:36.730
So the probability of not
getting it on any one day

00:01:36.730 --> 00:01:39.540
is four out of five,

00:01:39.540 --> 00:01:42.600
and the probability of not
getting it on two of the days,

00:01:42.600 --> 00:01:43.640
I would just multiply them

00:01:43.640 --> 00:01:45.810
because they are independent events.

00:01:45.810 --> 00:01:49.400
So four over five times four over five.

00:01:49.400 --> 00:01:52.210
So the probability that X is equal to 0

00:01:52.210 --> 00:01:57.210
is going to be 16/25, 16 over 25.

00:01:57.490 --> 00:02:02.490
Now what about the probability
that X is equal to 1?

00:02:03.090 --> 00:02:05.040
What is this going to be?

00:02:05.040 --> 00:02:08.290
Well, there are two scenarios over here.

00:02:08.290 --> 00:02:12.670
There's one scenario
where let's say on day one

00:02:12.670 --> 00:02:14.370
he does not get the dessert,

00:02:14.370 --> 00:02:17.940
and on day two he does get the dessert.

00:02:17.940 --> 00:02:20.490
But then of course
there's the other scenario

00:02:20.490 --> 00:02:23.430
where on day one he gets the dessert,

00:02:23.430 --> 00:02:28.070
and then on day two he
doesn't get the dessert.

00:02:28.070 --> 00:02:29.670
These are the two scenarios

00:02:29.670 --> 00:02:31.770
where he's going to get X equals 1.

00:02:31.770 --> 00:02:34.280
And so if we add these
together, let's see.

00:02:34.280 --> 00:02:39.095
4/5 times 1/5, this is going to be 4/25,

00:02:39.095 --> 00:02:43.200
and then this is going to be 4/25 again.

00:02:43.200 --> 00:02:45.070
And you add these two together,

00:02:45.070 --> 00:02:48.890
you're going to get 8/25.

00:02:48.890 --> 00:02:50.050
And then last but not least,

00:02:50.050 --> 00:02:51.670
and actually we could
figure out this last one

00:02:51.670 --> 00:02:55.270
by subtracting 16 and 8 from 25,

00:02:55.270 --> 00:02:57.180
which would actually give us 1/25th,

00:02:57.180 --> 00:02:58.450
but let's just write this out.

00:02:58.450 --> 00:03:01.480
The probability that X equals 2,

00:03:01.480 --> 00:03:03.710
this is the probability that
he gets a dessert on both days.

00:03:03.710 --> 00:03:06.790
So 1/5 chance on day one

00:03:06.790 --> 00:03:09.080
and 1/5 chance on the second day.

00:03:09.080 --> 00:03:12.730
So 1/5 times 1/5 is 1/25.

00:03:12.730 --> 00:03:14.380
And you can do a reality check here.

00:03:14.380 --> 00:03:16.090
These all need to add up to one.

00:03:16.090 --> 00:03:17.830
And they do indeed add up to one.

00:03:17.830 --> 00:03:20.396
16 plus 8 plus 1 is 25.

00:03:20.396 --> 00:03:22.940
So 25/25 is what they all add up to.

00:03:22.940 --> 00:03:23.773
And we're done.


Theoretical probability distribution example: tables

WEBVTT
Kind: captions
Language: en

00:00:00.360 --> 00:00:01.560
- [Instructor] We're
told that a board game

00:00:01.560 --> 00:00:05.450
has players roll two
3-sided dice, these exist

00:00:05.450 --> 00:00:07.030
and actually I looked it up, they do exist

00:00:07.030 --> 00:00:08.280
and they're actually fascinating.

00:00:08.280 --> 00:00:12.120
And subtract the numbers
showing on the faces.

00:00:12.120 --> 00:00:16.290
The game only looks at
non-negative differences.

00:00:16.290 --> 00:00:20.490
For example, if a player
rolls a one and a three,

00:00:20.490 --> 00:00:22.660
the difference is two.

00:00:22.660 --> 00:00:26.690
Let D represent the
difference in a given roll.

00:00:26.690 --> 00:00:30.760
Construct the theoretical
probability distribution of D.

00:00:30.760 --> 00:00:32.840
So pause this video and see
if you can have a go at that

00:00:32.840 --> 00:00:35.410
before we work through it together.

00:00:35.410 --> 00:00:38.520
All right, now let's
work through it together.

00:00:38.520 --> 00:00:40.630
So let's just think about
all of the scenarios

00:00:40.630 --> 00:00:42.370
for the two die.

00:00:42.370 --> 00:00:44.560
So let me draw a little table here.

00:00:44.560 --> 00:00:46.220
So let me do it like that

00:00:46.220 --> 00:00:49.250
and let me do it like this.

00:00:49.250 --> 00:00:52.870
And then let me put a little
divider right over here.

00:00:52.870 --> 00:00:56.790
And for this top, this
is going to be die one

00:00:56.790 --> 00:00:59.010
and then this is going to be die two.

00:00:59.010 --> 00:01:03.660
Die one can take on one, two, or three

00:01:03.660 --> 00:01:08.250
and die two could be one, two, or three.

00:01:08.250 --> 00:01:12.103
And so let me finish making
this a bit of a table here.

00:01:13.850 --> 00:01:16.180
And what we wanna do is
look at the difference

00:01:16.180 --> 00:01:17.890
but the non-negative difference.

00:01:17.890 --> 00:01:21.480
So we'll always subtract the
lower die from the higher die.

00:01:21.480 --> 00:01:22.870
So what's the difference here?

00:01:22.870 --> 00:01:24.460
Well, this is going to be zero.

00:01:24.460 --> 00:01:26.590
If I roll a one and a one.

00:01:26.590 --> 00:01:28.420
Now, what if I roll a two and a one?

00:01:28.420 --> 00:01:30.990
Well, here the difference is
going to be two minus one,

00:01:30.990 --> 00:01:32.040
which is one.

00:01:32.040 --> 00:01:35.630
Here the difference is three
minus one, which is two.

00:01:35.630 --> 00:01:37.450
Now what about right over here?

00:01:37.450 --> 00:01:41.370
Well, here the higher die
is two the lower one is one,

00:01:41.370 --> 00:01:42.230
right over here.

00:01:42.230 --> 00:01:45.600
So two minus one is one,

00:01:45.600 --> 00:01:47.900
two minus two is zero.

00:01:47.900 --> 00:01:50.240
And now this is gonna be the higher roll,

00:01:50.240 --> 00:01:52.670
die one is gonna have the
high roll in this scenario.

00:01:52.670 --> 00:01:54.570
Three minus two is one.

00:01:54.570 --> 00:01:59.080
And then right over here,
three minus one is two.

00:01:59.080 --> 00:02:00.960
Now die one rolls a two,

00:02:00.960 --> 00:02:03.090
die two rolls a three.

00:02:03.090 --> 00:02:05.960
Die three is higher,
three minus two is one.

00:02:05.960 --> 00:02:08.350
And then three minus three is zero.

00:02:08.350 --> 00:02:10.470
So we've come up with all of the scenarios

00:02:10.470 --> 00:02:13.440
and we can see that we're
either gonna end up with a zero

00:02:13.440 --> 00:02:16.900
or one or a two when we look
at the positive difference.

00:02:16.900 --> 00:02:21.900
So there's a scenario of
getting a zero, a one or a two.

00:02:22.330 --> 00:02:24.570
Those are the different differences

00:02:24.570 --> 00:02:26.140
that we could actually get.

00:02:26.140 --> 00:02:28.780
And so let's think about the
probability of each of them.

00:02:28.780 --> 00:02:31.780
What's the probability that
the difference is zero.

00:02:31.780 --> 00:02:36.010
Well, we can see that one, two, three

00:02:36.010 --> 00:02:38.940
of the nine equally likely outcomes,

00:02:38.940 --> 00:02:40.580
result in a difference of zero.

00:02:40.580 --> 00:02:44.670
So it's gonna be three
out of nine or one-third.

00:02:44.670 --> 00:02:49.670
What about a difference of,
let me use the blue, one?

00:02:49.900 --> 00:02:54.300
Well, we could see there
are one, two, three, four

00:02:54.300 --> 00:02:56.450
of the nine scenarios have that.

00:02:56.450 --> 00:02:59.000
So there is a four ninths probability.

00:02:59.000 --> 00:03:02.070
And then last but not
least a difference of two.

00:03:02.070 --> 00:03:05.470
Well, there's two out of the
nine scenarios that have that.

00:03:05.470 --> 00:03:09.408
So there is a two ninths
probability right over there.

00:03:09.408 --> 00:03:10.390
And we're done.

00:03:10.390 --> 00:03:13.090
We've constructed the theoretical
probability distribution

00:03:13.090 --> 00:03:13.953
of D.


Interpreting general multiplication rule

WEBVTT
Kind: captions
Language: en

00:00:00.230 --> 00:00:01.610
- [Instructor] We're
told that two contestants

00:00:01.610 --> 00:00:03.990
are finalists in a cooking competition.

00:00:03.990 --> 00:00:06.530
For the final round,
each of them spin a wheel

00:00:06.530 --> 00:00:09.950
to determine what star
ingredient must be in their dish.

00:00:09.950 --> 00:00:11.420
I guess the primary ingredient,

00:00:11.420 --> 00:00:12.820
and we can see it could be chard, spinach,

00:00:12.820 --> 00:00:16.720
romaine lettuce, I'm guessing,
cabbage, arugula, or kale.

00:00:16.720 --> 00:00:19.510
And so then they give us these
different types of events,

00:00:19.510 --> 00:00:21.860
or at least the symbols for
these different types of events,

00:00:21.860 --> 00:00:23.040
and then give us their meaning.

00:00:23.040 --> 00:00:27.800
So K-sub 1 means, the first
contestant lands on kale,

00:00:27.800 --> 00:00:32.800
K-sub 2 means, the second
contestant lands on kale,

00:00:32.800 --> 00:00:36.030
K-sub 1 with this superscript C,

00:00:36.030 --> 00:00:37.730
which you could view as complement.

00:00:37.730 --> 00:00:39.480
So K-sub 1 one complement,

00:00:39.480 --> 00:00:44.350
the first contestant
does not land on kale.

00:00:44.350 --> 00:00:46.970
So it's the complement of
this one right over here.

00:00:46.970 --> 00:00:49.010
And then K-sub 2 complement,

00:00:49.010 --> 00:00:52.670
would be that the second
contestant does not land on kale.

00:00:52.670 --> 00:00:56.440
So the not of K-sub 2 right over here.

00:00:56.440 --> 00:00:59.570
Using the general multiplication rule,

00:00:59.570 --> 00:01:02.530
express symbolically the probability

00:01:02.530 --> 00:01:06.700
that neither contestant lands on kale.

00:01:06.700 --> 00:01:09.550
So pause this video and see
if you can have a go at this.

00:01:10.480 --> 00:01:14.760
All right, so the general
multiplication rule

00:01:14.760 --> 00:01:16.610
is just saying this notion

00:01:16.610 --> 00:01:21.610
that the probability
of two events, A and B,

00:01:22.810 --> 00:01:26.740
is going to be equal
to the probability of,

00:01:26.740 --> 00:01:31.740
let's say A given B, times
the probability of B.

00:01:33.020 --> 00:01:34.820
Now, if they're independent events,

00:01:34.820 --> 00:01:36.740
if the probability of A occurring

00:01:36.740 --> 00:01:40.930
does not depend in any way
on whether B occurred or not,

00:01:40.930 --> 00:01:45.160
then this would simplify to
this probability of A given B,

00:01:45.160 --> 00:01:47.100
would just become the probability of A.

00:01:47.100 --> 00:01:48.850
And so if you have two independent events,

00:01:48.850 --> 00:01:50.570
you would just multiply their probability.

00:01:50.570 --> 00:01:51.940
So that's just all they're talking about,

00:01:51.940 --> 00:01:53.870
the general multiplication rule.

00:01:53.870 --> 00:01:54.860
But let me express

00:01:54.860 --> 00:01:57.270
what they're actually
asking us to express.

00:01:57.270 --> 00:02:01.770
The probability that neither
contestant lands on kale.

00:02:01.770 --> 00:02:03.790
So that means that this
is going to happen,

00:02:03.790 --> 00:02:06.360
the first contestant
does not land on kale,

00:02:06.360 --> 00:02:07.800
and this is going to happen,

00:02:07.800 --> 00:02:10.990
the second contestant
does not land on kale.

00:02:10.990 --> 00:02:12.630
So I could write it this way.

00:02:12.630 --> 00:02:17.320
The probability that K-sub 1 complement

00:02:17.320 --> 00:02:22.320
and K-sub 2 complement, and
I could write it this way.

00:02:24.470 --> 00:02:25.960
This is going to be equal to,

00:02:25.960 --> 00:02:28.650
we know that these are independent events

00:02:28.650 --> 00:02:31.530
because if the first contestant gets kale

00:02:31.530 --> 00:02:32.740
or whatever they get it,

00:02:32.740 --> 00:02:34.600
it doesn't get taken out of the running

00:02:34.600 --> 00:02:35.730
for the second contestant.

00:02:35.730 --> 00:02:38.410
The second contestant still
has an equal probability

00:02:38.410 --> 00:02:40.080
of getting or not getting kale,

00:02:40.080 --> 00:02:42.780
regardless of what happened
for the first contestant.

00:02:42.780 --> 00:02:44.290
So that means we're just in the situation

00:02:44.290 --> 00:02:45.930
where we multiply these probabilities.

00:02:45.930 --> 00:02:50.930
So that's gonna be the
probability of K-sub 1 complement,

00:02:51.060 --> 00:02:56.060
times the probability
of K-sub 2 complement.

00:02:56.560 --> 00:02:58.570
All right, now let's do part two.

00:02:58.570 --> 00:03:00.380
Interpret what each part

00:03:00.380 --> 00:03:03.860
of this probability statement represents.

00:03:03.860 --> 00:03:05.010
So I encourage you like always,

00:03:05.010 --> 00:03:07.210
pause this video and
try to figure that out.

00:03:08.090 --> 00:03:11.950
All right, so first let's think
about what is going on here.

00:03:11.950 --> 00:03:13.340
So this is saying,

00:03:13.340 --> 00:03:18.340
the probability that this
is K-sub 1 complement.

00:03:19.130 --> 00:03:22.600
So the first contestant
does not land on kale.

00:03:22.600 --> 00:03:27.600
So first, first contestant
does not get kale,

00:03:33.790 --> 00:03:36.920
and, I'll write and in caps,

00:03:36.920 --> 00:03:39.600
and second contestant does get kale.

00:03:39.600 --> 00:03:44.130
And second does get kale.

00:03:45.490 --> 00:03:47.670
So that's what this left-hand is saying.

00:03:47.670 --> 00:03:51.190
And now they say that that
is going to be equal to,

00:03:51.190 --> 00:03:52.330
so this part right over here,

00:03:52.330 --> 00:03:57.330
probability that the first
contestant does not get kale.

00:03:58.270 --> 00:04:03.270
Probability that first does not get kale,

00:04:06.930 --> 00:04:10.440
times, right over here.

00:04:10.440 --> 00:04:12.450
And the second part right over here

00:04:12.450 --> 00:04:17.450
is the probability that the
second contestant gets kale,

00:04:18.600 --> 00:04:22.170
given that the first
contestant does not get kale.

00:04:22.170 --> 00:04:27.130
So probability that the second gets kale,

00:04:29.130 --> 00:04:32.920
given, that's what this vertical
line right over here means,

00:04:32.920 --> 00:04:35.110
it means given, shorthand for given.

00:04:35.110 --> 00:04:36.780
Given, I wrote it up there too.

00:04:36.780 --> 00:04:41.780
Given that first does not get kale.

00:04:46.350 --> 00:04:49.300
And we're done, we've just
explained what is going on here.


General multiplication rule example: independent events

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:01.310
- [Instructor] We're
told that Maya and Doug

00:00:01.310 --> 00:00:03.870
are finalists in a crafting competition.

00:00:03.870 --> 00:00:06.400
For the final round,
each of them spin a wheel

00:00:06.400 --> 00:00:10.510
to determine what star material
must be in their craft.

00:00:10.510 --> 00:00:14.640
Maya and Doug both want to get
silk as their star material.

00:00:14.640 --> 00:00:17.990
Maya will spin first, followed by Doug.

00:00:17.990 --> 00:00:22.410
What is the probability that
neither contestant gets silk?

00:00:22.410 --> 00:00:24.600
Pause this video and think
through this on your own

00:00:24.600 --> 00:00:26.450
before we work through this together.

00:00:27.630 --> 00:00:30.960
All right, so first let's think
about what they're asking.

00:00:30.960 --> 00:00:33.890
They want to figure out the
probability that neither

00:00:33.890 --> 00:00:35.970
gets silk, so I'm gonna
write this in shorthand.

00:00:35.970 --> 00:00:40.723
So I'm going to use MNS for Maya no silk.

00:00:41.610 --> 00:00:45.260
And we're also thinking about Doug

00:00:45.260 --> 00:00:47.990
not being able to pick silk.

00:00:47.990 --> 00:00:51.810
So Maya no silk and Doug no silk.

00:00:51.810 --> 00:00:54.200
So we know that this could be viewed

00:00:54.200 --> 00:00:59.200
as the probability that
Maya doesn't get silk.

00:00:59.420 --> 00:01:02.740
She, after all does get
to spin this wheel first,

00:01:02.740 --> 00:01:04.350
and then we can multiply that

00:01:04.350 --> 00:01:09.350
by the probability that
Doug doesn't get silk,

00:01:09.350 --> 00:01:13.860
Doug no silk, given that
Maya did not get silk.

00:01:13.860 --> 00:01:16.030
Maya no silk.

00:01:16.030 --> 00:01:19.390
Now it's important to think
about whether Doug's probability

00:01:19.390 --> 00:01:24.032
is independent or dependent on
whether Maya got silk or not.

00:01:24.032 --> 00:01:28.060
So let's remember Maya will
spin first, but it's not like

00:01:28.060 --> 00:01:30.430
if she picks silk, that somehow silk

00:01:30.430 --> 00:01:31.890
is taken out of the running.

00:01:31.890 --> 00:01:33.160
In fact, no matter what she picks,

00:01:33.160 --> 00:01:34.660
it's not taken out of the running.

00:01:34.660 --> 00:01:36.580
Doug will then spin it again.

00:01:36.580 --> 00:01:39.230
And so these are really
two independent events,

00:01:39.230 --> 00:01:42.090
and so the probability
that Doug doesn't get silk

00:01:42.090 --> 00:01:44.160
given that Maya doesn't get silk,

00:01:44.160 --> 00:01:45.400
this is going to be the same thing

00:01:45.400 --> 00:01:48.790
as the probability that
just Doug doesn't get silk.

00:01:48.790 --> 00:01:51.550
It doesn't matter what happens to Maya.

00:01:51.550 --> 00:01:53.930
And so what are each of these?

00:01:53.930 --> 00:01:55.300
Well, this is all going to be equal

00:01:55.300 --> 00:01:57.850
to the probability that
Maya does not get silk.

00:01:57.850 --> 00:02:00.630
There's six pieces or six options

00:02:00.630 --> 00:02:02.330
of this wheel right over here.

00:02:02.330 --> 00:02:06.730
Five of them entail her not
getting silk on her spin.

00:02:06.730 --> 00:02:09.080
So five over six.

00:02:09.080 --> 00:02:12.350
And then similarly, when
Doug goes to spin this wheel

00:02:12.350 --> 00:02:13.870
there are six possibilities.

00:02:13.870 --> 00:02:17.480
Five of them are showing
that he does not get silk,

00:02:17.480 --> 00:02:18.680
Doug no silk.

00:02:18.680 --> 00:02:23.480
So times 5/6, which is of
course going to be equal

00:02:23.480 --> 00:02:27.423
to 25/36, and we're done.


General multiplication rule example: dependent events

WEBVTT
Kind: captions
Language: en

00:00:00.330 --> 00:00:02.390
- [Instructor] We're told that
Maya and Doug are finalists

00:00:02.390 --> 00:00:04.170
in a crafting competition.

00:00:04.170 --> 00:00:06.080
For the final round, each of them

00:00:06.080 --> 00:00:08.705
will randomly select a
card without replacement

00:00:08.705 --> 00:00:11.560
that will reveal what the star material

00:00:11.560 --> 00:00:13.460
must be in their craft.

00:00:13.460 --> 00:00:15.480
Here are the available cards.

00:00:15.480 --> 00:00:17.870
So I guess the star material
is the primary material

00:00:17.870 --> 00:00:20.540
they need to use in this competition.

00:00:20.540 --> 00:00:25.540
Maya and Doug both want to get
silk as their star material.

00:00:25.810 --> 00:00:29.630
Maya will draw first, followed by Doug.

00:00:29.630 --> 00:00:34.570
What is the probability that
neither contestant draws silk?

00:00:34.570 --> 00:00:36.480
Pause this video and see if
you can work through that

00:00:36.480 --> 00:00:38.330
before we work through this together.

00:00:39.980 --> 00:00:41.940
All right, now let's work
through this together.

00:00:41.940 --> 00:00:45.730
So the probably that neither
contestant draws silk.

00:00:45.730 --> 00:00:47.490
So that would be, I'll
just write it another way,

00:00:47.490 --> 00:00:52.310
the probability that, I'll
write MNS for Maya no silk.

00:00:52.310 --> 00:00:57.310
So Maya no silk and Doug no silk.

00:01:00.190 --> 00:01:02.029
That's just another way of
saying, what is the probability

00:01:02.029 --> 00:01:04.740
that neither contestant draws silk?

00:01:04.740 --> 00:01:09.540
And so this is going to be
equivalent to the probability

00:01:09.540 --> 00:01:14.540
that Maya does not get silk,
Maya no silk, right over here,

00:01:15.010 --> 00:01:20.010
times the probability that
Doug doesn't get silk,

00:01:20.800 --> 00:01:24.160
given that Maya did not get silk.

00:01:24.160 --> 00:01:26.060
Given Maya no silk.

00:01:26.060 --> 00:01:28.340
This line right over, this vertical line,

00:01:28.340 --> 00:01:30.560
this is shorthand for given.

00:01:30.560 --> 00:01:32.840
And so let's calculate each of these.

00:01:32.840 --> 00:01:35.020
So this is going to be
equal to the probability

00:01:35.020 --> 00:01:36.420
that Maya gets no silk.

00:01:36.420 --> 00:01:40.030
She picked first there's
six options out of here.

00:01:40.030 --> 00:01:44.680
Five of them are not silk,
so it is five over six.

00:01:44.680 --> 00:01:47.030
And then the probability
that Doug does not get silk,

00:01:47.030 --> 00:01:49.690
given that Maya did not get silk.

00:01:49.690 --> 00:01:52.212
So Maya did not get silk, then that means

00:01:52.212 --> 00:01:55.070
that silk is still in the mix,

00:01:55.070 --> 00:01:57.490
but there's only five possibilities left

00:01:57.490 --> 00:01:59.290
because Maya picked one of them,

00:01:59.290 --> 00:02:01.810
and four of them are not silk.

00:02:01.810 --> 00:02:03.800
There's still silk as an option.

00:02:03.800 --> 00:02:05.900
And it's important to
recognize that the probability

00:02:05.900 --> 00:02:09.130
that Doug gets no silk is dependent

00:02:09.130 --> 00:02:12.080
on whether Maya got silk or not.

00:02:12.080 --> 00:02:14.950
So it's very important to have
this given right over here.

00:02:14.950 --> 00:02:16.930
If these were independent events,

00:02:16.930 --> 00:02:19.770
if Maya picked and then
put her card back in

00:02:19.770 --> 00:02:21.840
and then Doug were to pick separately,

00:02:21.840 --> 00:02:24.080
then the probability
that Doug gets no silk,

00:02:24.080 --> 00:02:26.620
given that Maya got no silk,
would be the same thing,

00:02:26.620 --> 00:02:29.438
as a probability that Doug
gets no silk regardless

00:02:29.438 --> 00:02:30.930
of what Maya was doing.

00:02:30.930 --> 00:02:35.220
And so this will end up
becoming four over six

00:02:35.220 --> 00:02:37.713
which is the same thing as two thirds.


Probability with combinations example: choosing cards

WEBVTT
Kind: captions
Language: en

00:00:00.160 --> 00:00:02.560
- We're told that a standard
deck of 52 playing cards

00:00:02.560 --> 00:00:06.700
includes four aces, four
kings, and 44 other cards.

00:00:06.700 --> 00:00:09.950
Suppose that Luis
randomly draws four cards

00:00:09.950 --> 00:00:11.940
without replacement.

00:00:11.940 --> 00:00:15.310
What is the probability
that Luis gets two aces

00:00:15.310 --> 00:00:18.490
and two kings, in any order?

00:00:18.490 --> 00:00:19.340
So like always,

00:00:19.340 --> 00:00:22.040
pause this video and see if
you can work through this.

00:00:23.110 --> 00:00:23.943
All right.

00:00:23.943 --> 00:00:25.410
Now, to figure out this probability,

00:00:25.410 --> 00:00:26.243
we can think about this,

00:00:26.243 --> 00:00:29.110
it's going to be the number of,

00:00:29.110 --> 00:00:31.430
let's call them draws

00:00:31.430 --> 00:00:36.200
with exactly two aces and two kings,

00:00:36.200 --> 00:00:41.200
two aces and two kings.

00:00:42.220 --> 00:00:43.053
And that's going to be

00:00:43.053 --> 00:00:47.700
over the total number of
possible draws of four cards.

00:00:47.700 --> 00:00:48.700
So number

00:00:49.560 --> 00:00:54.130
of possible draws

00:00:55.760 --> 00:00:59.300
of four cards.

00:00:59.300 --> 00:01:00.300
Now, for many of y'all,

00:01:00.300 --> 00:01:01.920
this bottom, the denominator here,

00:01:01.920 --> 00:01:03.980
might be a little bit
easier to think about.

00:01:03.980 --> 00:01:06.760
We know that there's 52 total cards,

00:01:06.760 --> 00:01:09.030
of which we are choosing four.

00:01:09.030 --> 00:01:11.330
So we could say 52 choose four,

00:01:11.330 --> 00:01:13.270
and that will tell us the total number

00:01:13.270 --> 00:01:15.610
of possible draws of four cards.

00:01:15.610 --> 00:01:17.700
How many combinations
of four cards can we get

00:01:17.700 --> 00:01:19.880
when we're picking from 52?

00:01:19.880 --> 00:01:23.053
But the top here might be a
little bit more of a stumper.

00:01:24.170 --> 00:01:28.040
We can think we have
exactly two spots for aces,

00:01:28.040 --> 00:01:30.260
so we're choosing two aces

00:01:30.260 --> 00:01:32.400
out of how many possible aces?

00:01:32.400 --> 00:01:34.120
Well, there's four total aces.

00:01:34.120 --> 00:01:38.020
So if we say four choose two,

00:01:38.020 --> 00:01:40.270
this is the total number of ways,

00:01:40.270 --> 00:01:41.500
when you don't care about order,

00:01:41.500 --> 00:01:45.040
that you can have two out
of your four aces picked.

00:01:45.040 --> 00:01:45.873
And then separately,

00:01:45.873 --> 00:01:47.690
we can use similar logic to say,

00:01:47.690 --> 00:01:52.260
all right, there's also
four choose two ways

00:01:52.260 --> 00:01:55.940
of picking two kings out
of four possible kings.

00:01:55.940 --> 00:01:57.870
And now the total number of draws

00:01:57.870 --> 00:02:00.080
with two aces and two kings,

00:02:00.080 --> 00:02:02.950
this is going to be the
product of these two.

00:02:02.950 --> 00:02:05.540
And if you're wondering why
you can just multiply it,

00:02:05.540 --> 00:02:06.430
think about it.

00:02:06.430 --> 00:02:09.970
For every scenario that
you have these two aces,

00:02:09.970 --> 00:02:12.720
you have four choose two scenarios

00:02:12.720 --> 00:02:15.230
of which kings you're dealing with.

00:02:15.230 --> 00:02:17.700
So you would take the product of them.

00:02:17.700 --> 00:02:19.520
And we've already done many examples

00:02:19.520 --> 00:02:21.610
of computing combinatorics like this,

00:02:21.610 --> 00:02:22.660
so I will leave you there.

00:02:22.660 --> 00:02:24.810
If you're so motivated,
I encourage you to be,

00:02:24.810 --> 00:02:27.533
you can actually calculate this value.


Probability with combinations example: choosing groups

WEBVTT
Kind: captions
Language: en

00:00:00.420 --> 00:00:01.410
- [Instructor] We're told that Kyra works

00:00:01.410 --> 00:00:03.930
on a team of 13 total people.

00:00:03.930 --> 00:00:06.470
Her manager is randomly
selecting three members

00:00:06.470 --> 00:00:09.760
from her team to represent
the company at a conference.

00:00:09.760 --> 00:00:12.430
What is the probability
that Kyra is chosen

00:00:12.430 --> 00:00:13.810
for the conference?

00:00:13.810 --> 00:00:15.360
Pause this video and
see if you can have a go

00:00:15.360 --> 00:00:18.270
at this before we work
through this together.

00:00:18.270 --> 00:00:20.250
All right, now let's work
through this together.

00:00:20.250 --> 00:00:22.640
So we wanna figure out this probability.

00:00:22.640 --> 00:00:24.600
And so one way to think about it is,

00:00:24.600 --> 00:00:27.440
what are the number of
ways that Kyra can be

00:00:27.440 --> 00:00:31.245
on a team or the number of possible teams,

00:00:31.245 --> 00:00:34.463
teams with Kyra,

00:00:36.390 --> 00:00:41.390
and then over the total
number of possible teams,

00:00:41.400 --> 00:00:46.203
total number of possible teams.

00:00:48.760 --> 00:00:52.200
And if this little hint
gets you even more inspired.

00:00:52.200 --> 00:00:53.500
If you weren't able to
do it the first time,

00:00:53.500 --> 00:00:55.230
I encourage you to try to pause it again

00:00:55.230 --> 00:00:56.980
and then work through it.

00:00:56.980 --> 00:00:59.740
All right, now I will
continue to continue.

00:00:59.740 --> 00:01:01.590
So first let me do the denominator here.

00:01:01.590 --> 00:01:04.650
What are the total
possible number of teams?

00:01:04.650 --> 00:01:05.660
Some of y'all might've found

00:01:05.660 --> 00:01:07.900
that a little bit easier to figure out.

00:01:07.900 --> 00:01:10.810
Well, we know that we're
choosing from 13 people

00:01:10.810 --> 00:01:12.880
and we're picking three of them

00:01:12.880 --> 00:01:14.700
and we don't care about order.

00:01:14.700 --> 00:01:17.180
It's not like we're saying
someone's gonna be president

00:01:17.180 --> 00:01:18.013
of the team,

00:01:18.013 --> 00:01:19.120
someone's gonna be vice-president

00:01:19.120 --> 00:01:20.580
and someone's gonna be treasurer.

00:01:20.580 --> 00:01:23.250
We just say there are
three people in the team.

00:01:23.250 --> 00:01:26.680
And so this is a
situation where out of 13,

00:01:26.680 --> 00:01:30.210
we are choosing three people.

00:01:30.210 --> 00:01:32.160
Now, what are the total number of teams,

00:01:32.160 --> 00:01:35.660
possible teams that could have Kyra in it?

00:01:35.660 --> 00:01:38.330
Well, one way to think
about it is if we know

00:01:38.330 --> 00:01:41.600
that Kyra is on a team,
then the possibilities are

00:01:41.600 --> 00:01:44.310
who's gonna be the other
two people on the team,

00:01:44.310 --> 00:01:45.920
and who are the possible candidates

00:01:45.920 --> 00:01:47.260
for the other two people?

00:01:47.260 --> 00:01:49.260
Well, if Kyra is already on the team

00:01:49.260 --> 00:01:51.970
then there's a possible
12 people to pick from.

00:01:51.970 --> 00:01:53.810
So there's 12 people to choose from

00:01:53.810 --> 00:01:55.680
for those other two slots.

00:01:55.680 --> 00:01:57.430
And so we're gonna choose two.

00:01:57.430 --> 00:01:58.510
And once again, we don't care

00:01:58.510 --> 00:02:00.920
about the order with which
we are choosing them.

00:02:00.920 --> 00:02:04.290
So once again, it is
gonna be a combination.

00:02:04.290 --> 00:02:05.960
And then we can just go ahead

00:02:05.960 --> 00:02:09.690
and calculate each of
these combinations here.

00:02:09.690 --> 00:02:11.890
What is 12 choose two?

00:02:11.890 --> 00:02:13.890
Well, there's 12 possible people

00:02:13.890 --> 00:02:16.370
for that first nine Kyra's seat.

00:02:16.370 --> 00:02:18.750
And then there would be 11 people there

00:02:18.750 --> 00:02:21.270
for that other non Kyra's spot.

00:02:21.270 --> 00:02:22.640
And of course it's a combination.

00:02:22.640 --> 00:02:24.550
We don't care what order we picked it in.

00:02:24.550 --> 00:02:27.580
And so there are two ways
to get these two people.

00:02:27.580 --> 00:02:28.750
We could say two factorial

00:02:28.750 --> 00:02:31.250
but that's just the same
thing as two or two times one.

00:02:31.250 --> 00:02:33.210
And then the denominator here.

00:02:33.210 --> 00:02:37.250
For that first spot, there's
13 people to pick from ,

00:02:37.250 --> 00:02:39.410
then in that second spot, there are 12.

00:02:39.410 --> 00:02:42.240
Then in that third spot, there are 11.

00:02:42.240 --> 00:02:45.210
And then once again, we
don't care about order,

00:02:45.210 --> 00:02:47.530
three factorial ways to
arrange three people.

00:02:47.530 --> 00:02:50.540
So I could write three times two,

00:02:50.540 --> 00:02:52.800
and for kicks I could
write one right over here,

00:02:52.800 --> 00:02:55.030
and then we can, let's go down here.

00:02:55.030 --> 00:02:57.150
This is gonna be equal to my numerator

00:02:57.150 --> 00:03:00.700
over here is gonna be six times 11.

00:03:00.700 --> 00:03:05.700
And then my denominator is
going to be 12 divided by six

00:03:06.560 --> 00:03:08.180
right over here is two.

00:03:08.180 --> 00:03:13.180
So it's gonna be 13 times 11 times two.

00:03:14.410 --> 00:03:16.700
Just to be clear, I divided
both the denominator

00:03:16.700 --> 00:03:18.520
and this numerator over here

00:03:18.520 --> 00:03:21.160
by six to get two right over there.

00:03:21.160 --> 00:03:23.530
Now this cancels with that.

00:03:23.530 --> 00:03:26.790
And then if we divide the
numerator and denominator by two,

00:03:26.790 --> 00:03:28.380
this is gonna be three here.

00:03:28.380 --> 00:03:29.580
This is gonna be one.

00:03:29.580 --> 00:03:33.120
And so we are left with a probability

00:03:33.120 --> 00:03:37.623
of 3/13 that Kyra is
chosen for the conference.