Probability from compound sample space

WEBVTT
Kind: captions
Language: en

00:00:00.345 --> 00:00:01.700
- [Voiceover] Let's say
that you're on some type

00:00:01.700 --> 00:00:04.069
of a game show and you've
been doing quite well.

00:00:04.069 --> 00:00:05.605
And you're now at the
round where you get to

00:00:05.605 --> 00:00:08.469
pick your fantabulous vacation.

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And so there are three possible
places that you could go.

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You could go on an island beach vacation.

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Island beach vacation.

00:00:20.670 --> 00:00:24.910
You could go skiing on a ski vacation.

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Or, you could go camping.

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Now those aren't the only possibilities

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because for each of those vacations

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there's different amount of
time that you could go on them.

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So you could go for one day.

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You could go for two days.

00:00:42.166 --> 00:00:43.301
Two days.

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Or you could go for three days.

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Three. Put that in a different color.

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You could go for three days.

00:00:52.603 --> 00:00:55.205
You could go for three days.

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So the first question I'd wanna
know is, well what is the--

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They're gonna randomly pick either a

00:01:02.245 --> 00:01:05.440
one day ski vacation or a
two day island vacation.

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The first question I wanna know is,

00:01:06.573 --> 00:01:09.013
what are all of the
possible outcomes here?

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What is the sample space?

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What is the space from
which we are going to pick

00:01:12.669 --> 00:01:15.546
your particular vacation package?

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Well for the sample space,
we can construct a grid.

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Which you can see that I've
essentially been constructing

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while I wrote down all
of the possibilities.

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So let me draw out the sample space

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with these uneven looking grid lines.

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All right, I think you get the picture.

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And I'll just abbreviate it.

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You can go on a one day,

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a one day island trip.

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This one I, this is one day island trip.

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You can go on a two day, two day--

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Actually, let me just write it this way.

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All of these are going to be one day.

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Right? Because of the one day column.

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All of these are going to be two days.

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Two days. Two days.

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And all of these are
going to be three days

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because it's on the three day column.

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And all of the ones in this row

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are going to be island trips.

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So it's one day island trip,

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two day island trip,
three day island trip.

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This second row, it's all ski trips.

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One day ski trip, two day
ski trip, three day ski trip.

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And then finally everything
in this third row,

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they're campng trips.

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One day camping trip,
two day camping trip,

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three day camping trip.

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So just like that, we have constructed

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the sample space right over here.

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You see that there's one,
two, three, four, five,

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six, seven, eight, nine outcomes.

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And let's say that each of these outcomes

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are on a little piece of paper

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and they put it in a
barrel and they roll it up.

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And for our purposes we can assume

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that they are all equally likely outcomes.

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So we're gonna assume
equally likely outcomes.

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So if we do assume
equally likely outcomes,

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we can figure out a probability.

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Maybe you live in someplace that's cold

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and you're really not in
the mood to go skiing.

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In fact, you'd like to spend several days

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away from the snow.

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So let's ask ourselves a question.

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What is the probability

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that you're going to win something

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at least two days

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on a vacation without snow.

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Two days on vacation

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without snow.

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You're gonna randomly pick
one of these nine outcomes.

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What's the probability that
it's going to be at least--

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It's gonna give you a
vacation that gives you

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at least two days without snow.

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Well, let's just think
a little bit about it.

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We know the sample space

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and we know each of the
outcomes are equally likely.

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There are nine equal outcomes here.

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So let's write that down.

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We got nine equal outcomes.

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Now how many of the outcomes satisfy this?

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This event, this constraint.

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At least 2 days of vacation--

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Let me write this.

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Two days.

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Without snow.

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Whether it falls or
touching it or whatever.

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So you're essentially avoiding skiing.

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You want at least two days on
something other than skiing.

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And we're assuming you're
not gonna go camping

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in some type of alpine.

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You're camping in some place that's warm.

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Well, let's think about these outcomes.

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So this one is no snow
but it's only one day.

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This is two days without snow
so we can circle that one.

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This is three days without
snow so we can circle that one.

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All of these have snow.

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This is one day without snow
so we're not gonna do this one.

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This is two days without snow

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and this is three days without snow.

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And so four of the equally

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likely outcomes satisfy this constraint.

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So you have a 4/9 probability

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of getting a vacation

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that keeps you away from
snow for at least two days.

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Hopefully you found that fun
and useful for the next time

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that you are some type
of strange game show.


Compound sample spaces

WEBVTT
Kind: captions
Language: en

00:00:00.418 --> 00:00:02.344
- [Voiceover] What I'm hoping
to explore in this video

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is the notion of a sample space.

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And it's a pretty, hopefully you'll find,

00:00:08.748 --> 00:00:10.119
straightforward idea.

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If you're doing a trial,

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something that is probabilistic,

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a trial or an experiment,

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a sample space is just the
set of the possible outcomes.

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So a very simple trial
might be a coin flip.

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So if you're talking about a coin flip,

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well then the sample space is going to be

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the set of all the possible outcomes.

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So you could get a heads

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or you could get a tails.

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That right over here is the sample space

00:00:41.161 --> 00:00:43.336
for the coin flip.

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And it's very useful because,

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for example, if these are
equally likely outcomes

00:00:48.701 --> 00:00:50.651
and you say well what's the probability

00:00:50.651 --> 00:00:51.966
of the event of a heads, you say

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okay that's one out of the
two equally likely outcomes.

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Or you could even construct, once you know

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all of the possible outcomes even if

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they aren't equally likely, you could say

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well let's create a
probability distribution.

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We at least know what the sample space is.

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We know what the possible outcomes are,

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now let's think about the probability

00:01:06.501 --> 00:01:07.871
of each of those outcomes.

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But a lot of times when people
talk about sample spaces,

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they're often, they tend to
be most useful, I would say,

00:01:14.117 --> 00:01:16.206
when you have equally likely outcomes

00:01:16.206 --> 00:01:19.086
like in the case of a fair coin flip.

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Because then from the sample space,

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it's fairly straight forward
to think about the probability

00:01:23.868 --> 00:01:26.028
of various events.

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But this is a simple sample
space right over here,

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but let's make things a
little bit more interesting.

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Let's imagine a world,

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so let's just put this aside a little bit.

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Let's imagine a world
where there's a bakery

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and at that bakery there are three types,

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three flavors of cupcakes,

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but there's also three
different sizes of cupcakes.

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So now we're essentially
looking at two different ways

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in which the thing that we're
going to be sampling can vary.

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So what we're doing,
let me write this down.

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So we have our flavors,

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flavors of cupcakes at this bakery.

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And let's say that you have chocolate,

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chocolate,

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you have, let's say there's strawberry,

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strawberry,

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and let's say that there is vanilla,

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there is vanilla.

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And they come in three different sizes.

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So sizes.

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Sizes could be small,

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I'll just write it out, small,

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medium,

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or large.

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So if you were, and let's
say each of these flavors

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come in each of these sizes.

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Or you could view it the other way around,

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each of these sizes come
in all three flavors.

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So now how do you
construct the sample space?

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If you were to say, look I'm gonna go,

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you know, I'm gonna blindfold myself

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and walk into this bakery and randomly

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somehow pickup a cupcake and my fingers

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can't tell the flavor or
the size of the cupcake,

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what are the possible,
what are the possible

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outcomes for the cupcake I'll pick?

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And the outcome would be both the flavor

00:03:02.358 --> 00:03:04.671
and the size of the cupcake.

00:03:05.393 --> 00:03:07.924
Well there's a bunch of
ways to think about this.

00:03:07.924 --> 00:03:10.548
One way is you could draw a tree.

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You could say, okay well I'm gonna pick

00:03:12.661 --> 00:03:13.891
three different flavors.

00:03:13.891 --> 00:03:15.418
I could either pick chocolate,

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chocolate,

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I'm gonna pick strawberry,

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strawberry,

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or I'm going to pick vanilla,

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vanilla.

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And then for each of those flavors,

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I'm only gonna pick a
small, medium, or large.

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So you could say small,

00:03:33.376 --> 00:03:35.650
medium, large.

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Small, so this is a small chocolate,

00:03:37.950 --> 00:03:40.364
this is a medium chocolate,
this is a large chocolate.

00:03:40.364 --> 00:03:44.056
This is a small strawberry,
medium strawberry,

00:03:44.056 --> 00:03:45.728
large strawberry.

00:03:45.728 --> 00:03:49.327
This is a small vanilla, medium vanilla,

00:03:49.327 --> 00:03:50.976
large vanilla.

00:03:50.976 --> 00:03:53.228
And so you see there's
nine possible outcomes.

00:03:53.228 --> 00:03:55.829
Once again, this is a medium chocolate.

00:03:55.829 --> 00:03:58.638
You picked a chocolate
and it was a medium one.

00:03:58.638 --> 00:04:00.611
This is a large vanilla.

00:04:00.611 --> 00:04:03.049
You picked a vanilla,
and it is a large one.

00:04:03.049 --> 00:04:04.257
And you could have done
it the other way around.

00:04:04.257 --> 00:04:05.324
You could have said, well okay,

00:04:05.324 --> 00:04:07.902
I'm going to either pick a small,

00:04:07.902 --> 00:04:10.224
medium, or large

00:04:10.224 --> 00:04:11.548
and then for each of those I'm gonna pick

00:04:11.548 --> 00:04:13.451
either a chocolate,
strawberry, or vanilla.

00:04:13.451 --> 00:04:15.299
And I'll just use the first letters.

00:04:15.299 --> 00:04:17.105
So I'm either gonna pick a chocolate,

00:04:17.105 --> 00:04:19.520
a strawberry, or a vanilla.

00:04:19.520 --> 00:04:22.237
When I write the S over
here in this magenta color,

00:04:22.237 --> 00:04:23.769
I'm talking about the flavor

00:04:23.769 --> 00:04:25.463
and if I write the S in green,

00:04:25.463 --> 00:04:27.228
I'm talking about small.

00:04:27.228 --> 00:04:30.689
So here you can have, if
you have a medium cupcake,

00:04:30.689 --> 00:04:32.035
it could be chocolate,

00:04:32.035 --> 00:04:33.312
it could be strawberry,

00:04:33.312 --> 00:04:34.515
or it could be vanilla.

00:04:34.515 --> 00:04:35.489
If you have a large cupcake

00:04:35.489 --> 00:04:37.023
it could be chocolate,

00:04:37.023 --> 00:04:38.346
strawberry,

00:04:38.346 --> 00:04:40.529
or vanilla.

00:04:40.529 --> 00:04:44.011
So for example, this was a
medium, chocolate cupcake.

00:04:44.011 --> 00:04:47.540
Over here a medium, chocolate
cupcake is this one.

00:04:47.540 --> 00:04:49.368
It's medium, chocolate.

00:04:50.065 --> 00:04:52.085
It would be that one over here.

00:04:52.085 --> 00:04:53.526
So you could use these

00:04:53.526 --> 00:04:55.523
kind of a tree diagram like this

00:04:55.523 --> 00:04:57.961
to think about the sample space,

00:04:57.961 --> 00:05:00.468
to think about the nine
possible outcomes here.

00:05:00.468 --> 00:05:01.746
But you could also,

00:05:01.746 --> 00:05:03.626
you could also do a,

00:05:03.626 --> 00:05:04.787
I guess you could say grid,

00:05:04.787 --> 00:05:05.965
where you could write

00:05:05.965 --> 00:05:07.979
the flavors, so you could have chocolate,

00:05:09.513 --> 00:05:10.806
chocolate,

00:05:11.829 --> 00:05:12.874
actually, let me just write the,

00:05:12.874 --> 00:05:14.337
well no, let me write them out.

00:05:14.337 --> 00:05:15.637
Actually, let me just,

00:05:15.637 --> 00:05:16.868
let me just write the letters since this

00:05:16.868 --> 00:05:18.424
is gonna take a long time to do.

00:05:18.424 --> 00:05:20.212
So you could have the flavors chocolate,

00:05:20.212 --> 00:05:21.813
strawberry, vanilla.

00:05:21.813 --> 00:05:24.205
So that's along that axis.

00:05:24.205 --> 00:05:25.785
And then you have your sizes.

00:05:25.785 --> 00:05:27.944
You can have a small,

00:05:27.944 --> 00:05:29.616
a medium,

00:05:29.616 --> 00:05:31.102
or large.

00:05:31.102 --> 00:05:33.023
And you can set up a grid here.

00:05:33.930 --> 00:05:36.687
So this is another way to do it.

00:05:36.687 --> 00:05:40.019
And notice this grid has nine boxes.

00:05:40.019 --> 00:05:41.332
So let's look at it.

00:05:42.309 --> 00:05:44.397
So set up the grid.

00:05:45.722 --> 00:05:47.279
Set up the grid.

00:05:47.279 --> 00:05:49.020
And so what is this one going to be?

00:05:49.020 --> 00:05:51.365
This is a small chocolate.

00:05:51.365 --> 00:05:52.426
Small.

00:05:53.334 --> 00:05:55.842
Small chocolate.

00:05:55.842 --> 00:05:57.861
Small chocolate.

00:05:57.861 --> 00:05:58.977
What is this one?

00:05:58.977 --> 00:06:01.826
This is going to be a small,

00:06:02.896 --> 00:06:04.560
a small strawberry,

00:06:05.353 --> 00:06:08.557
A small strawberry.

00:06:08.557 --> 00:06:10.507
And you could just keep
constructing like this

00:06:10.507 --> 00:06:13.604
where everything in this row,
there are all about small.

00:06:13.604 --> 00:06:14.259
Oops.

00:06:14.259 --> 00:06:15.857
Do it in the...

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This is small.

00:06:17.780 --> 00:06:19.753
I'm having trouble changing colors.

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Alright (laughing).

00:06:21.262 --> 00:06:25.558
There is a small and then this is a small,

00:06:25.558 --> 00:06:27.625
this is a small vanilla.

00:06:27.625 --> 00:06:29.690
This color changing is really,

00:06:29.690 --> 00:06:31.037
it's a difficult thing.

00:06:31.037 --> 00:06:32.454
Small vanilla.

00:06:32.454 --> 00:06:33.871
And all of these, these are all,

00:06:33.871 --> 00:06:35.751
this would be a medium chocolate.

00:06:35.751 --> 00:06:37.238
Medium strawberry.

00:06:37.238 --> 00:06:38.560
Medium vanilla.

00:06:38.560 --> 00:06:41.300
Large chocolate, large strawberry,

00:06:41.300 --> 00:06:42.206
large vanilla.

00:06:42.206 --> 00:06:43.785
And once again you have nine outcomes.

00:06:43.785 --> 00:06:45.086
This is another way to think about

00:06:45.086 --> 00:06:47.245
all of the possible outcomes when

00:06:47.245 --> 00:06:49.126
you're looking at these two ways

00:06:49.126 --> 00:06:51.609
in which my cupcakes could vary.

00:06:51.609 --> 00:06:53.235
Another way, a third way
that you could do it,

00:06:53.235 --> 00:06:54.815
is you could literally
just construct a table.

00:06:54.815 --> 00:06:56.625
Well you could say, okay

00:06:56.625 --> 00:06:58.993
I could have a chocolate,

00:06:58.993 --> 00:07:00.340
actually I'm going to
use the letters again.

00:07:00.340 --> 00:07:04.776
So let's say we make,
this is the flavor column.

00:07:04.776 --> 00:07:07.671
And then this is the size column.

00:07:08.857 --> 00:07:10.087
Size column.

00:07:10.087 --> 00:07:11.412
And so you could say

00:07:11.412 --> 00:07:14.709
I could have a chocolate that is,

00:07:14.709 --> 00:07:16.520
so let's see, there's
three types of chocolate

00:07:16.520 --> 00:07:17.553
that I could have.

00:07:18.396 --> 00:07:20.272
And they could be

00:07:20.272 --> 00:07:24.172
they could be small, medium, or large.

00:07:24.172 --> 00:07:27.068
You could say there's three types of,

00:07:27.678 --> 00:07:29.531
three types of strawberry.

00:07:29.531 --> 00:07:31.782
It could be small, medium, or large.

00:07:31.782 --> 00:07:33.263
So let me write that in.

00:07:33.263 --> 00:07:36.119
Small, medium, large.

00:07:36.119 --> 00:07:38.755
Or you could say oh, there's
three types of vanilla.

00:07:39.997 --> 00:07:42.472
There's three types of,

00:07:42.472 --> 00:07:44.237
I'm color changing again.

00:07:44.237 --> 00:07:46.188
Three types of vanilla.

00:07:46.188 --> 00:07:47.929
Once again, it could be

00:07:47.929 --> 00:07:50.738
small, medium, or large.

00:07:50.738 --> 00:07:53.386
So you have these nine possibilities.

00:07:53.386 --> 00:07:55.753
Now, sample space,

00:07:55.753 --> 00:07:57.077
the sample space isn't telling you

00:07:57.077 --> 00:07:59.121
if they're equally likely or not.

00:07:59.121 --> 00:08:00.114
It's just telling you, hey, if

00:08:00.114 --> 00:08:01.321
you're gonna do an experiment,

00:08:01.321 --> 00:08:03.365
what are all the different possibilities,

00:08:03.365 --> 00:08:06.521
the possible outcomes for that experiment?

00:08:06.521 --> 00:08:08.844
Now in the case where
they are equally likely,

00:08:08.844 --> 00:08:11.770
it can be very, very
useful because you can say

00:08:11.770 --> 00:08:14.323
you could do something like, if you said

00:08:14.323 --> 00:08:16.367
okay, it's equally likely to pick any one

00:08:16.367 --> 00:08:18.386
of these nine outcomes, you could say

00:08:18.386 --> 00:08:20.825
well what's the probability of,

00:08:20.825 --> 00:08:23.867
what's the probability of
getting something that is

00:08:23.867 --> 00:08:27.025
either small or chocolate?

00:08:27.025 --> 00:08:28.743
And so you could see well
how many of those events

00:08:28.743 --> 00:08:31.436
out of the total actually
meet that constraint?

00:08:31.436 --> 00:08:33.526
But we'll do more of
that in future videos.

00:08:33.526 --> 00:08:35.732
That's just a little bit of a clue

00:08:35.732 --> 00:08:38.170
of why we even care about things

00:08:38.170 --> 00:08:40.283
like sample spaces.

00:08:40.283 --> 00:08:42.372
Especially sample spaces like this,

00:08:42.372 --> 00:08:44.230
where we're looking along two ways

00:08:44.230 --> 00:08:46.947
or multiple ways that something can vary.

00:08:46.947 --> 00:08:49.292
And these types of sample
spaces in particular

00:08:49.292 --> 00:08:52.356
are called compound sample spaces.

00:08:52.356 --> 00:08:53.913
So these right over here,

00:08:53.913 --> 00:08:57.163
this is a compound sample space,

00:08:57.163 --> 00:08:58.975
because we're looking
at two different ways

00:08:58.975 --> 00:08:59.856
that it can vary.

00:08:59.856 --> 00:09:01.064
Not just a heads or tails,

00:09:01.064 --> 00:09:03.177
it can vary by size or by flavor.

00:09:03.177 --> 00:09:04.896
And you can even have compound

00:09:04.896 --> 00:09:07.587
sample spaces that vary
in more than two ways.


Constructing probability model from observations

WEBVTT
Kind: captions
Language: en

00:00:00.751 --> 00:00:03.457
- [Voiceover] Let's say
that you love frozen yogurt.

00:00:03.457 --> 00:00:05.477
So every day after school you decide

00:00:05.477 --> 00:00:07.230
to go to the frozen yogurt store

00:00:07.230 --> 00:00:11.154
at exactly four o'clock, four o'clock PM.

00:00:11.897 --> 00:00:13.871
Now, because you like
frozen yogurt so much,

00:00:13.871 --> 00:00:16.089
you are not a big fan of having to wait

00:00:16.089 --> 00:00:18.573
in line when you get
there, you're impatient,

00:00:18.573 --> 00:00:21.092
you want your frozen yogurt immediately.

00:00:21.092 --> 00:00:23.227
And so you decide to conduct a study.

00:00:23.227 --> 00:00:24.749
You want to figure out the probability

00:00:24.749 --> 00:00:26.815
of there being lines of different sizes

00:00:26.815 --> 00:00:29.091
when you go to the frozen yogurt store

00:00:29.091 --> 00:00:32.388
after school, exactly at four o'clock PM.

00:00:32.388 --> 00:00:35.035
So in your study, the
next 50 times you observe,

00:00:35.035 --> 00:00:37.194
you go to the frozen
yogurt store at four PM,

00:00:37.194 --> 00:00:39.214
you make a series of observations.

00:00:39.214 --> 00:00:41.930
You observe the size of the line.

00:00:41.930 --> 00:00:43.753
So, let me make two columns here,

00:00:43.753 --> 00:00:48.506
line size is the left column,

00:00:48.506 --> 00:00:50.274
and on the right column,

00:00:50.274 --> 00:00:53.906
let's say this is the
number of times observed.

00:00:53.906 --> 00:00:58.322
So, times observed, observed.

00:01:01.463 --> 00:01:05.897
All right, times observed,
my handwriting is,

00:01:05.897 --> 00:01:10.548
O-B-S-E-R-V-E-D, all
right, times observed.

00:01:11.459 --> 00:01:13.287
All right, so let's first think about it.

00:01:13.287 --> 00:01:15.168
Okay, so you go and you say, hey look,

00:01:15.168 --> 00:01:19.185
I see no people in line,
exactly, or you see

00:01:19.185 --> 00:01:22.593
no people in line, exactly 24 times.

00:01:23.257 --> 00:01:27.194
You see one person in line

00:01:27.194 --> 00:01:29.448
exactly 18 times,

00:01:30.438 --> 00:01:35.069
and you see two people in
line exactly eight times.

00:01:35.069 --> 00:01:37.757
And, in your 50 visits,
you don't see more,

00:01:37.757 --> 00:01:39.638
you never see more than
two people in line.

00:01:39.638 --> 00:01:42.157
I guess this is a very efficient cashier

00:01:42.157 --> 00:01:45.141
at this frozen yogurt store.

00:01:45.141 --> 00:01:47.996
So based on this, based
on what you have observed,

00:01:47.996 --> 00:01:52.196
what would be your estimate
of the probabilities

00:01:52.196 --> 00:01:55.431
of finding no people in
line, one people in line,

00:01:55.431 --> 00:01:59.170
or two people in line,
at four PM on the days

00:01:59.170 --> 00:02:01.688
after school that you visit
the frozen yogurt store?

00:02:01.688 --> 00:02:04.591
You only visit it on weekdays
where there are schooldays.

00:02:04.591 --> 00:02:06.204
So what's the probability
of there being no line,

00:02:06.204 --> 00:02:08.259
a one person line, or a two person line

00:02:08.259 --> 00:02:11.239
when you visit at four PM on a school day?

00:02:12.308 --> 00:02:16.545
Well, all you can do is
estimate the true probability,

00:02:16.545 --> 00:02:18.265
the true theoretical probability.

00:02:18.265 --> 00:02:19.808
We don't know what that is, but

00:02:19.808 --> 00:02:22.201
you've done 50 observations here right.

00:02:23.354 --> 00:02:25.187
I know that this adds up to 50,

00:02:25.187 --> 00:02:29.180
18 plus eight is 26, 26 plus 24 is 50,

00:02:29.180 --> 00:02:30.830
so you've done 50 observations here

00:02:30.830 --> 00:02:32.432
and so you can figure out, well what are

00:02:32.432 --> 00:02:34.905
the relative frequencies
of having zero people?

00:02:34.905 --> 00:02:37.064
What is the relative
frequency of one person,

00:02:37.064 --> 00:02:40.291
or the relative frequency
of two people in line?

00:02:40.291 --> 00:02:41.429
And then we can use that as

00:02:41.429 --> 00:02:43.309
the estimates for the probability.

00:02:43.309 --> 00:02:44.072
So let's do that.

00:02:44.909 --> 00:02:47.401
So, probability estimate.

00:02:47.401 --> 00:02:49.433
I'll do it in the next column.

00:02:49.433 --> 00:02:53.223
So probability, probability estimate,

00:02:54.803 --> 00:02:56.684
and once again we can do that by

00:02:56.684 --> 00:02:58.704
looking at the relative frequency.

00:02:58.704 --> 00:03:02.001
The relative frequency of zero, well we

00:03:02.001 --> 00:03:05.821
observed that 24 times out of 50.

00:03:05.821 --> 00:03:08.808
So, 24 out of 50 is the same thing

00:03:08.808 --> 00:03:13.011
as 0.48,

00:03:13.011 --> 00:03:16.446
or you could even say that this is 48%.

00:03:17.630 --> 00:03:19.294
Now, what's the relative frequency

00:03:19.294 --> 00:03:20.794
of seeing one person in line?

00:03:21.780 --> 00:03:24.495
Well you observed that
18 out of the 50 visits,

00:03:25.682 --> 00:03:28.433
18 out of the 50 visits, that
would be a relative frequency,

00:03:28.433 --> 00:03:31.795
18 divided by 50 is 0.36,

00:03:31.795 --> 00:03:35.926
which is 36% of your visits.

00:03:35.926 --> 00:03:38.520
And then, finally, the relative frequency

00:03:38.520 --> 00:03:40.691
of seeing a two person line, that was

00:03:40.691 --> 00:03:42.977
eight out of the 50 visits,

00:03:43.759 --> 00:03:47.467
and so that it 0.16,

00:03:47.467 --> 00:03:51.364
and that is equal to 16% of the visits.

00:03:51.975 --> 00:03:53.502
And so, there's interesting things here.

00:03:53.502 --> 00:03:55.220
Remember, these are
estimates of the probability.

00:03:55.220 --> 00:03:57.902
You're doing this by essentially sampling

00:03:57.902 --> 00:04:00.328
what the line on 50 different
days, you don't know,

00:04:00.328 --> 00:04:02.848
it's not gonna always be exactly this,

00:04:02.848 --> 00:04:05.906
but it's a good estimate,
you did it 50 times.

00:04:06.722 --> 00:04:08.358
And so based on this, you'd
say, well I'd estimate

00:04:08.358 --> 00:04:11.458
the probability of having
a zero person line as 48%.

00:04:11.458 --> 00:04:13.629
I'd estimate the probability of

00:04:13.629 --> 00:04:15.731
having a one person line as 36%.

00:04:15.731 --> 00:04:17.692
I'd estimate the probability of having

00:04:17.692 --> 00:04:21.013
a two person line is 16%, or is 0.16.

00:04:21.013 --> 00:04:22.533
It's important to realize that

00:04:22.533 --> 00:04:23.775
these are legitimate probabilities.

00:04:23.775 --> 00:04:25.737
Remember, to be a
probability, it has to be

00:04:25.737 --> 00:04:29.058
between zero and one, it
has to be zero and one.

00:04:29.058 --> 00:04:30.949
And if you look at all
of the possible events,

00:04:30.949 --> 00:04:33.886
it should add up to one,
because at least based on

00:04:33.886 --> 00:04:36.034
your observations, these
are the possibilities.

00:04:36.034 --> 00:04:38.205
Obviously in a real world,
it might be some kind

00:04:38.205 --> 00:04:39.865
of crazy thing where
more people go in line.

00:04:39.865 --> 00:04:42.489
But at least based on the
events that you've seen,

00:04:42.489 --> 00:04:44.253
these three different
events, and these are

00:04:44.253 --> 00:04:46.053
the only three that you've observed,

00:04:46.053 --> 00:04:50.371
based on your observations,
these three should add,

00:04:50.371 --> 00:04:51.926
cause these are the only
three things you've observed,

00:04:51.926 --> 00:04:54.863
they should add up to one,
and they do add up to one.

00:04:54.863 --> 00:04:57.873
Let's see, 36 plus 16 is 52,

00:04:57.873 --> 00:05:00.961
52 plus 48, they add up to one.

00:05:00.961 --> 00:05:02.743
Now, if once you do this,

00:05:02.743 --> 00:05:04.745
you might do something interesting.

00:05:04.745 --> 00:05:06.035
You might say, okay, you know what,

00:05:06.035 --> 00:05:09.317
over the next two years,

00:05:09.317 --> 00:05:13.139
you plan on visiting 500 times.

00:05:13.139 --> 00:05:17.967
So visiting 500 times,

00:05:20.797 --> 00:05:22.253
so based on your estimates

00:05:22.253 --> 00:05:24.993
of the probability of having no line,

00:05:24.993 --> 00:05:27.325
of a one person line,
or a two person line,

00:05:27.325 --> 00:05:31.070
how many times in your
next 500 visits would you

00:05:31.070 --> 00:05:34.448
expect there to be a two person line?

00:05:34.448 --> 00:05:37.246
Based on your observations so far.

00:05:37.246 --> 00:05:39.847
Well, it's reasonable to say,

00:05:39.847 --> 00:05:44.386
well a good estimate
of the number of times

00:05:44.386 --> 00:05:47.142
you'll see a two person line
when you visit 500 times,

00:05:47.142 --> 00:05:49.857
well you say, well there's
gonna be 500 times,

00:05:49.857 --> 00:05:52.574
and it's a reasonable expectation,

00:05:52.574 --> 00:05:55.140
based on your estimate
of the probability that

00:05:55.140 --> 00:05:57.309
0.16 of the time,

00:05:57.309 --> 00:06:01.370
you will see a two person line.

00:06:01.370 --> 00:06:03.309
Or you could say eight
out of every 50 times.

00:06:04.345 --> 00:06:06.608
And so what is this going to be?

00:06:06.608 --> 00:06:09.508
Let's see, 500 divided by 50 is just 10,

00:06:09.508 --> 00:06:14.508
so you would expect that
80 out of the 500 times

00:06:14.510 --> 00:06:18.692
you would see a two person line.

00:06:18.692 --> 00:06:20.680
Now to be clear, I would be shocked if

00:06:20.680 --> 00:06:22.550
it's exactly 80 ends up being the case,

00:06:22.550 --> 00:06:23.930
but this is actually a very good

00:06:23.930 --> 00:06:26.183
expectation based on your observations.

00:06:26.183 --> 00:06:28.527
It is completely possible, first of all,

00:06:28.527 --> 00:06:30.443
that your observations were off.

00:06:30.443 --> 00:06:33.485
That you, you know, that
it's just the random chance

00:06:33.485 --> 00:06:35.400
that you happened to observe this many

00:06:35.400 --> 00:06:38.083
or this few times that there
were two people in line.

00:06:38.083 --> 00:06:39.127
So that could be off.

00:06:39.127 --> 00:06:41.891
But even if these are very good estimates,

00:06:41.891 --> 00:06:45.884
it's possible that something, that you see

00:06:45.884 --> 00:06:48.753
a two person line 85 out of the 500 times,

00:06:48.753 --> 00:06:51.655
or 65 out of the 500 times.

00:06:51.655 --> 00:06:53.489
All of those things are possible.

00:06:53.489 --> 00:06:56.112
And it's always very
important to keep in mind,

00:06:56.112 --> 00:06:59.828
you're estimating the
true probability here,

00:06:59.828 --> 00:07:01.639
which it's very hard to know for sure

00:07:01.639 --> 00:07:04.158
what the true probability is,
but you can make estimates

00:07:04.158 --> 00:07:07.521
based on sampling the
line on different days,

00:07:07.521 --> 00:07:11.352
by making these observations,
by having these experiments

00:07:11.352 --> 00:07:13.300
so to speak, each of these observations

00:07:13.300 --> 00:07:14.728
you could use in experiment, and then you

00:07:14.728 --> 00:07:17.061
can use those to set an expectation.

00:07:17.061 --> 00:07:19.941
But none of these things
do you know for sure,

00:07:19.941 --> 00:07:21.020
that they're definitely gonna be

00:07:21.020 --> 00:07:23.363
exactly 80 out of the next 500 times.


Comparing theoretical to experimental probabilites

WEBVTT
Kind: captions
Language: en

00:00:00.351 --> 00:00:02.963
- Let's say that you've got a bag,

00:00:02.963 --> 00:00:06.177
and in that bag you
put a bunch of marbles.

00:00:06.177 --> 00:00:08.980
So, let's say you put 50
of these magenta marbles.

00:00:08.980 --> 00:00:11.773
So one, two, three,
four, five, six, seven,

00:00:11.773 --> 00:00:14.756
I'm not gonna draw all of them
but you get the general idea.

00:00:14.756 --> 00:00:18.862
There are going to be 50 magenta marbles,

00:00:21.502 --> 00:00:24.850
and there's also going
to be 50 blue marbles.

00:00:31.020 --> 00:00:34.700
And what you do is, you have
these 100 marbles in there,

00:00:34.700 --> 00:00:36.871
half of them magenta, half of them blue.

00:00:36.871 --> 00:00:38.531
And before picking a marble out,

00:00:38.531 --> 00:00:39.599
and you're gonna be blindfolded

00:00:39.599 --> 00:00:41.915
when you pick a marble out, you shake

00:00:41.915 --> 00:00:45.996
the bag really good to, so you think,

00:00:45.996 --> 00:00:47.702
mix them up a little bit.

00:00:47.702 --> 00:00:50.546
And so, if you were to say theoretically,

00:00:50.546 --> 00:00:52.729
what is the probability, if you

00:00:52.729 --> 00:00:54.645
stuck your hand in and you're not looking,

00:00:54.645 --> 00:00:59.645
what is the probability
of picking a magenta?

00:01:00.368 --> 00:01:03.433
I feel the need to write
magenta in magenta.

00:01:03.433 --> 00:01:08.433
What is the probability of
picking a magenta marble?

00:01:08.732 --> 00:01:11.164
Well, theoretically there's 100

00:01:11.164 --> 00:01:12.767
equally likely possibilities,

00:01:12.767 --> 00:01:15.270
there's 100 marbles in the bag.

00:01:15.270 --> 00:01:18.770
And 50 of them involve picking a magenta.

00:01:18.770 --> 00:01:20.811
So, 50 out of 100, when this is

00:01:20.811 --> 00:01:23.007
the same thing as a 1/2 probability.

00:01:23.007 --> 00:01:24.654
So you could say, well, "Theoretically,

00:01:24.654 --> 00:01:26.662
"there is a 1/2 probability,
I just did the math."

00:01:26.662 --> 00:01:30.014
If you say these are 100
equally likely possibilities,

00:01:30.014 --> 00:01:32.242
50 of them are picking magenta.

00:01:32.242 --> 00:01:34.719
Now let's say you actually
start doing the experiment.

00:01:34.719 --> 00:01:36.400
So you literally take a bag with

00:01:36.400 --> 00:01:38.970
50 magenta marbles, 50 blue marbles,

00:01:38.970 --> 00:01:43.662
and then you start picking the marbles,

00:01:43.662 --> 00:01:45.771
and then you see what
marble color you picked,

00:01:45.771 --> 00:01:48.169
and you put it back in,
and then you do it again.

00:01:48.169 --> 00:01:52.458
And so, let's say that after every time

00:01:52.458 --> 00:01:54.002
you put your hand in the bag and

00:01:54.002 --> 00:01:55.952
you take something out of the bag,

00:01:55.952 --> 00:01:57.380
and you observe what it is, we're gonna

00:01:57.380 --> 00:01:58.877
call that an experiment.

00:01:58.877 --> 00:02:03.877
So, after 10 experiments,

00:02:10.330 --> 00:02:13.118
let's say that you you have picked out

00:02:13.118 --> 00:02:18.118
seven magenta and three blue.

00:02:23.518 --> 00:02:28.518
So, is this strange that out
of the first 10 experiments,

00:02:29.364 --> 00:02:32.835
you haven't picked out exactly
half of them being magenta,

00:02:32.835 --> 00:02:34.467
you've picked out seven magenta,

00:02:34.467 --> 00:02:36.543
and then the other three were blue.

00:02:36.543 --> 00:02:38.747
Well no, this is definitely
a reasonable thing.

00:02:38.747 --> 00:02:41.977
If the true probability of
picking out a magenta is 1/2,

00:02:41.977 --> 00:02:43.960
it's definitely possible
that you could still

00:02:43.960 --> 00:02:46.127
pick out seven magenta, that just

00:02:46.127 --> 00:02:48.897
happened to be what your fingers touched.

00:02:48.897 --> 00:02:50.348
And this isn't a lot of experiments,

00:02:50.348 --> 00:02:51.973
it's completely reasonable that out of 10,

00:02:51.973 --> 00:02:56.199
yeah, you could have,
later on in statistics

00:02:56.199 --> 00:02:59.008
we'll define these things in more detail,

00:02:59.008 --> 00:03:01.744
but there's enough variation in where

00:03:01.744 --> 00:03:04.155
you might pick that you're
not going to always get,

00:03:04.155 --> 00:03:06.282
especially with only 10 experiments,

00:03:06.282 --> 00:03:09.900
you're not definitely
going to get exactly 1/2.

00:03:09.900 --> 00:03:11.425
Instead of having five magenta,

00:03:11.425 --> 00:03:13.906
it's completely reasonable
to have seven magenta.

00:03:13.906 --> 00:03:16.510
So, this really wouldn't
cause me a lot of pause.

00:03:16.510 --> 00:03:19.795
I still wouldn't question what I did here

00:03:19.795 --> 00:03:23.398
when I calculated this
theoretical probability.

00:03:23.398 --> 00:03:27.359
But let's say you have a
lot of time on your hands.

00:03:27.359 --> 00:03:32.359
And let's say after 10,000 trials here,

00:03:33.905 --> 00:03:38.265
after 10,000 experiments,
and remember the experiment;

00:03:38.265 --> 00:03:41.310
you're sticking your hand
in the bag without looking,

00:03:41.310 --> 00:03:44.668
your fingers kind of feeling
around, picks out a marble,

00:03:44.668 --> 00:03:48.435
and you observe the marble
and you record what you found.

00:03:48.435 --> 00:03:51.328
And so, let's say after
10,000 experiments,

00:03:51.328 --> 00:03:56.322
you get 7,000 magenta.

00:03:56.322 --> 00:03:58.003
Actually I'm gonna do
slightly different numbers,

00:03:58.003 --> 00:03:59.464
so let me make it even more extreme.

00:03:59.464 --> 00:04:04.304
Let's say you get 8,000 magenta

00:04:05.580 --> 00:04:08.691
and you have 2,000 blue.

00:04:13.598 --> 00:04:15.552
Now this is interesting, because here

00:04:15.552 --> 00:04:17.330
what you're seeing experimentally

00:04:17.330 --> 00:04:18.955
seems to be very different.

00:04:18.955 --> 00:04:21.218
And now you have a large number

00:04:21.219 --> 00:04:23.504
of trials right over here, not just 10.

00:04:23.504 --> 00:04:26.518
10 is completely reasonable that,

00:04:26.518 --> 00:04:27.662
hey you know, I got seven magenta and

00:04:27.662 --> 00:04:29.605
three blue instead of five and five,

00:04:29.605 --> 00:04:33.821
but now you've done 10,000.

00:04:33.821 --> 00:04:34.836
You would've expected if this was

00:04:34.836 --> 00:04:37.088
the true probability,
you would've expected

00:04:37.088 --> 00:04:39.398
that half of these would've been magenta,

00:04:39.398 --> 00:04:41.992
only 5,000 magenta and 5,000 blue,

00:04:41.992 --> 00:04:43.951
but you got 8,000 magenta.

00:04:43.951 --> 00:04:46.434
Now, this is within the
realm of possibility

00:04:46.434 --> 00:04:49.761
if the true probability of
picking a magenta is 1/2,

00:04:49.761 --> 00:04:52.924
but it's very unlikely
that you would've gotten

00:04:52.924 --> 00:04:54.775
this result with this many experiments,

00:04:54.775 --> 00:04:57.550
this many trials if the
true probability was 1/2.

00:04:57.550 --> 00:05:01.549
Here your experimental
probability is showing,

00:05:01.549 --> 00:05:05.185
look, out of 10,000 trials,

00:05:08.893 --> 00:05:13.140
experimental probability here is

00:05:13.140 --> 00:05:17.901
you had 10,000 trials,
or 10,000 experiments

00:05:17.901 --> 00:05:19.305
I guess you could say.

00:05:19.305 --> 00:05:24.305
And and in 8,000 of them,
you got a magenta marble.

00:05:26.286 --> 00:05:30.389
And so, this is going to be 80%, or 8/10.

00:05:32.587 --> 00:05:34.958
So, there seems to be a difference here.

00:05:34.958 --> 00:05:37.615
The reason why I would
take this more seriously

00:05:37.615 --> 00:05:39.844
is that you had a lot of trials here,

00:05:39.844 --> 00:05:43.189
you did this 10,000 times.

00:05:43.189 --> 00:05:45.219
If the true probability was one half,

00:05:45.219 --> 00:05:47.146
it's very low likelihood that you

00:05:47.146 --> 00:05:49.384
would've gotten this many magenta.

00:05:49.384 --> 00:05:50.687
So, when you think about it you're like,

00:05:50.687 --> 00:05:53.113
"What's going on here, what are

00:05:53.113 --> 00:05:55.040
"possible explanations for this?"

00:05:55.040 --> 00:05:56.189
This, I wouldn't have fretted about,

00:05:56.189 --> 00:05:58.232
after 10 experiments, not a big deal.

00:05:58.232 --> 00:06:01.473
But after 10,000, this
would have caused me pause.

00:06:01.473 --> 00:06:02.760
Well, why would this happen,

00:06:02.760 --> 00:06:04.130
I mixed up the bag every time.

00:06:04.130 --> 00:06:06.045
And there're some different possibilities;

00:06:06.045 --> 00:06:09.412
maybe the blue marbles
are slightly heavier,

00:06:09.412 --> 00:06:11.794
and so when you shake the bag up enough,

00:06:11.794 --> 00:06:13.847
the blue marbles settled to the bottom,

00:06:13.847 --> 00:06:17.462
and you're more likely
to pick a magenta marble.

00:06:17.462 --> 00:06:20.190
Maybe the blue marbles
have a slightly different

00:06:20.190 --> 00:06:23.768
texture to them, in which case, maybe

00:06:23.768 --> 00:06:26.187
they slip out of your hands,
or they're less likely

00:06:26.187 --> 00:06:27.551
to be gripped on, and so you're

00:06:27.551 --> 00:06:29.606
more likely to pick a magenta.

00:06:29.606 --> 00:06:31.649
So, I don't know the explanation,

00:06:31.649 --> 00:06:33.101
I don't know what's going on in that bag,

00:06:33.101 --> 00:06:35.318
but if I thought theoretically

00:06:35.318 --> 00:06:37.198
that the probability should be 1/2,

00:06:37.198 --> 00:06:39.335
because half of the marbles are magenta,

00:06:39.335 --> 00:06:41.076
but I'm seeing through my experiments

00:06:41.076 --> 00:06:43.038
that 80% of what I'm picking out,

00:06:43.038 --> 00:06:44.710
especially if I did 10,000 of them,

00:06:44.710 --> 00:06:47.008
if I did this 10,000 times, well,

00:06:47.008 --> 00:06:49.485
this is going to cause me some pause.

00:06:49.485 --> 00:06:50.769
I would really start to think about

00:06:50.769 --> 00:06:53.660
whether it's truly equally likely for me

00:06:53.660 --> 00:06:56.562
to pick out a red, a
magenta versus a blue.

00:06:56.562 --> 00:06:59.207
Something else must be going on.


Making predictions with probability

WEBVTT
Kind: captions
Language: en

00:00:00.558 --> 00:00:01.476
- [Voiceover] So right over here,

00:00:01.476 --> 00:00:04.540
I have a type of spinner
that you might see

00:00:04.540 --> 00:00:06.593
in a child's game.

00:00:06.593 --> 00:00:09.659
You can see there's seven
equally likely possibilities.

00:00:09.659 --> 00:00:10.999
Let's see, there's one, two,

00:00:10.999 --> 00:00:13.936
three, four, five, six,

00:00:13.936 --> 00:00:16.165
seven equally likely possibilities.

00:00:16.165 --> 00:00:17.442
It looks like in four of them,

00:00:17.442 --> 00:00:19.932
you've spun an elephant.

00:00:19.932 --> 00:00:21.147
Two of them, you have this mouse

00:00:21.147 --> 00:00:23.004
running away from something,

00:00:23.004 --> 00:00:23.650
and then in one of them,

00:00:23.650 --> 00:00:25.681
you have this monkey doing some type

00:00:25.681 --> 00:00:29.083
of acrobatics, fair enough.

00:00:29.083 --> 00:00:32.481
Now let's pose ourselves
an interesting question.

00:00:32.481 --> 00:00:34.541
At least I think it's
an interesting question.

00:00:34.541 --> 00:00:38.592
Let's say we were to spin
the spinner 210 times.

00:00:38.592 --> 00:00:42.517
So we're going to spin 210...

00:00:42.517 --> 00:00:44.154
We're going to spin that spinner

00:00:44.154 --> 00:00:46.416
210 times.

00:00:46.416 --> 00:00:48.657
And I want you to make a prediction.

00:00:48.657 --> 00:00:50.932
I want you to predict...

00:00:50.932 --> 00:00:55.926
I want you to predict
the number of times...

00:00:55.926 --> 00:01:00.115
Number of times...

00:01:00.582 --> 00:01:03.089
We get an elephant.

00:01:03.255 --> 00:01:05.740
We get an elephant...

00:01:05.740 --> 00:01:08.859
Number of times we get an elephant,

00:01:08.859 --> 00:01:11.261
out of the 210 times.

00:01:11.261 --> 00:01:13.503
So why don't you to have a go at it?

00:01:13.503 --> 00:01:16.318
All right, so you've, I'm
assuming, like always,

00:01:16.318 --> 00:01:18.149
pause the video and then had a try.

00:01:18.149 --> 00:01:19.226
So one way to think about it,

00:01:19.226 --> 00:01:20.142
is, well, for one spin,

00:01:20.142 --> 00:01:22.638
what is the probability
of getting an elephant?

00:01:22.638 --> 00:01:25.679
So let's do this, one spin.

00:01:25.679 --> 00:01:28.117
So for one spin,

00:01:28.117 --> 00:01:31.391
what is the probability of getting

00:01:31.391 --> 00:01:34.021
an elephant?

00:01:34.161 --> 00:01:35.992
Well let's see, we have
already talked about,

00:01:35.992 --> 00:01:37.448
this is a fair spinner.

00:01:37.448 --> 00:01:40.653
There are seven equally
likely possibilities.

00:01:40.653 --> 00:01:42.796
And then how many involve
getting an elephant?

00:01:42.796 --> 00:01:46.753
Well, so we have one, two, three, four.

00:01:46.753 --> 00:01:50.446
Four out of the seven
equally likely possibilities

00:01:50.446 --> 00:01:53.290
involve us getting an elephant.

00:01:53.290 --> 00:01:54.833
So one reasonable thing to do,

00:01:54.833 --> 00:01:56.134
and this is actually what I would do,

00:01:56.134 --> 00:01:59.966
is go, look, a 4/7 probability means

00:01:59.966 --> 00:02:03.490
I should expect that 4/7 of the time,

00:02:03.490 --> 00:02:05.839
especially if I'm doing it over,

00:02:05.839 --> 00:02:06.738
and over, and over again,

00:02:06.738 --> 00:02:09.229
it's a reasonable expectation that, hey,

00:02:09.229 --> 00:02:12.809
4/7 of the time, I will get an elephant.

00:02:12.809 --> 00:02:14.923
I've just calculated the theoretical

00:02:14.923 --> 00:02:16.909
probability here, based on this being

00:02:16.909 --> 00:02:18.092
a fair spinner.

00:02:18.092 --> 00:02:19.091
And that should inform,

00:02:19.091 --> 00:02:20.810
that if I were to do a
bunch of experiments,

00:02:20.810 --> 00:02:24.479
that 4/7 of the time that I should see

00:02:24.479 --> 00:02:26.666
me getting the elephant.

00:02:26.666 --> 00:02:29.245
So it would be a reasonable prediction

00:02:29.245 --> 00:02:31.368
to say well, look, I'm
going to spin this thing

00:02:31.368 --> 00:02:34.642
210 times, and I would expect

00:02:34.642 --> 00:02:37.660
that 4/7 of those 210 times,

00:02:37.660 --> 00:02:40.748
I would get an elephant.

00:02:40.748 --> 00:02:42.954
And so, let's think about what this is.

00:02:42.954 --> 00:02:45.090
210 times 4/7,

00:02:45.090 --> 00:02:47.600
210 divided by seven is 30.

00:02:47.600 --> 00:02:50.138
30 times 40 is 120.

00:02:50.138 --> 00:02:52.216
So 120 times.

00:02:52.216 --> 00:02:54.050
My prediction, or maybe your prediction

00:02:54.050 --> 00:02:55.849
was this as well, I think it's

00:02:55.849 --> 00:02:57.138
a reasonable prediction,

00:02:57.138 --> 00:03:00.947
is that if I spin it 210 times,

00:03:00.947 --> 00:03:04.186
that I'm going to get
an elephant 120 times.

00:03:04.186 --> 00:03:06.393
That's very important to think about,

00:03:06.393 --> 00:03:09.248
what this is saying,
and this is not saying.

00:03:09.248 --> 00:03:11.605
Is it possible that I get an elephant

00:03:11.605 --> 00:03:15.284
121 times, or maybe 119 times?

00:03:15.284 --> 00:03:17.804
Sure, sure, it's completely reasonable

00:03:17.804 --> 00:03:19.382
that you might get something
different than this.

00:03:19.382 --> 00:03:21.588
In fact, there's some probability

00:03:21.588 --> 00:03:23.321
that you get no elephant.

00:03:23.321 --> 00:03:24.967
If you consider getting an elephant lucky,

00:03:24.967 --> 00:03:28.124
that you just happen to keep landing

00:03:28.124 --> 00:03:29.421
on the monkey or one of the mice,

00:03:29.421 --> 00:03:30.919
and that's a very low probability

00:03:30.919 --> 00:03:33.682
that that would happen
if you spun it 210 times,

00:03:33.682 --> 00:03:36.503
but it is possible.

00:03:36.503 --> 00:03:37.699
So it's important to realize

00:03:37.699 --> 00:03:38.743
that this is just a prediction.

00:03:38.743 --> 00:03:40.481
There's actually a
possibility that you might get

00:03:40.481 --> 00:03:42.824
an elephant on all 210 spins.

00:03:42.824 --> 00:03:44.935
Once again, that's a low probability,

00:03:44.935 --> 00:03:46.234
but it is possible.

00:03:46.234 --> 00:03:47.930
So this isn't saying
that you're definitely

00:03:47.930 --> 00:03:52.019
going to get the elephant 120 times.

00:03:52.019 --> 00:03:54.003
In fact, it's very reasonable,

00:03:54.003 --> 00:03:56.580
that you might get the elephant 123 times,

00:03:56.580 --> 00:03:59.599
or 128 times, or 110 times,

00:03:59.599 --> 00:04:01.144
or even 90 times.

00:04:01.144 --> 00:04:05.399
These are all completely
reasonable things to happen.

00:04:05.399 --> 00:04:06.495
All you would say,

00:04:06.495 --> 00:04:08.399
is that, look, if I had to predict it,

00:04:08.399 --> 00:04:10.581
this, out of all of the different...

00:04:10.581 --> 00:04:13.018
I can get the elephant anywhere between

00:04:13.018 --> 00:04:16.239
zero and 210 times, out of all

00:04:16.239 --> 00:04:19.594
of those possibilities,
before I even start spinning,

00:04:19.594 --> 00:04:23.429
I'll say, okay, I think that this is the

00:04:23.429 --> 00:04:24.786
most reasonable one, that I'm going

00:04:24.786 --> 00:04:27.618
to get it 4/7 of the time.

00:04:27.618 --> 00:04:31.081
But it's not saying that, hey,

00:04:31.081 --> 00:04:33.827
that I'm definitely going
to get it 120 times,

00:04:33.827 --> 00:04:37.136
it's not saying that 118 times,

00:04:37.136 --> 00:04:41.687
or 129 times aren't reasonably

00:04:41.687 --> 00:04:43.568
possible as well.

00:04:43.568 --> 00:04:44.311
It's just saying, look,

00:04:44.311 --> 00:04:45.634
this is a reasonable prediction.

00:04:45.634 --> 00:04:48.119
I'm using the experimental probability,

00:04:48.119 --> 00:04:50.011
4/7 probability, and so,

00:04:50.011 --> 00:04:52.356
if I'm going to do something 210 times,

00:04:52.356 --> 00:04:53.981
well, I could expect that it's going

00:04:53.981 --> 00:04:55.954
to happen 4/7 of the time.

00:04:55.954 --> 00:04:57.231
I don't know for sure that it's going

00:04:57.231 --> 00:04:59.020
to happen 4/7 of the time,

00:04:59.020 --> 00:05:03.881
but that is a reasonable
prediction to make.


Intuitive sense of probabilities

WEBVTT
Kind: captions
Language: en

00:00:00.400 --> 00:00:01.700
- [Voiceover] What I
hope to do in this video

00:00:01.700 --> 00:00:05.600
is give ourselves a more
intuitive sense of probabilities.

00:00:05.600 --> 00:00:08.200
Let's go back to an example
that we've seen before.

00:00:08.200 --> 00:00:10.434
We're rolling a fair six-sided die.

00:00:10.434 --> 00:00:12.333
There are six possibilities.

00:00:12.333 --> 00:00:15.666
We could get a one, a two, a three,

00:00:15.666 --> 00:00:19.268
a four, a five or a six.

00:00:19.268 --> 00:00:22.100
Now let's say we ask ourselves

00:00:22.100 --> 00:00:23.800
what is the probability

00:00:23.800 --> 00:00:28.325
of rolling a number that is

00:00:28.325 --> 00:00:32.607
less than or equal to two?

00:00:32.607 --> 00:00:34.023
What is this going to be?

00:00:34.869 --> 00:00:38.067
Well, there are six equally
likely possibilities.

00:00:38.067 --> 00:00:40.133
Rolling less than or equal to two,

00:00:40.133 --> 00:00:43.253
well, that means I'm either
rolling a one or a two.

00:00:43.253 --> 00:00:46.038
So two, one, two, of the six

00:00:46.038 --> 00:00:49.400
equally likely possibilities
meet my constraints.

00:00:49.400 --> 00:00:52.667
So there is a 2/6 probability
of rolling a number

00:00:52.667 --> 00:00:54.567
less than or equal to two.

00:00:54.567 --> 00:00:56.812
Or I can just rewrite that
as an equivalent fraction.

00:00:56.812 --> 00:00:59.169
I could say there's a 1/3 probability.

00:00:59.169 --> 00:01:00.892
I could go either way.

00:01:00.892 --> 00:01:02.927
Now let's also ask
ourselves another question.

00:01:02.927 --> 00:01:07.866
What is the probability
of rolling a number

00:01:10.112 --> 00:01:13.570
greater than or equal to three?

00:01:13.586 --> 00:01:17.653
Once again, there are six
equally likely possibilities.

00:01:17.653 --> 00:01:19.319
How many of them involve rolling

00:01:19.319 --> 00:01:21.066
greater than or equal to three?

00:01:21.066 --> 00:01:23.299
Let's see, one, two, three, four,

00:01:23.299 --> 00:01:25.715
these possibilities right over here.

00:01:25.715 --> 00:01:28.490
Throw a three, a four, a five or six.

00:01:28.490 --> 00:01:32.300
So four out of the six
equally likely possibilities.

00:01:32.300 --> 00:01:34.902
Or I could rewrite this
as an equivalent fraction,

00:01:34.902 --> 00:01:36.933
as 2/3.

00:01:36.933 --> 00:01:38.666
So what's more likely?

00:01:38.666 --> 00:01:41.792
Rolling a number that's
less than or equal to two?

00:01:41.792 --> 00:01:44.538
Or rolling a number that's
greater than or equal to three?

00:01:44.538 --> 00:01:46.666
Well, you can see it right over here.

00:01:46.666 --> 00:01:48.867
The probability of rolling
greater than or equal to three

00:01:48.867 --> 00:01:51.266
is 2/3 while the probability
of rolling less than

00:01:51.266 --> 00:01:54.067
or equal to two is only 1/3.

00:01:54.067 --> 00:01:55.933
This number is greater.

00:01:55.933 --> 00:01:57.933
So this has a greater probability

00:01:57.933 --> 00:01:59.500
or another way of thinking about it,

00:01:59.500 --> 00:02:01.520
rolling greater than or equal to three

00:02:01.520 --> 00:02:05.786
is more likely than rolling
less than or equal to two.

00:02:05.786 --> 00:02:07.200
In fact, not only is it more likely,

00:02:07.200 --> 00:02:09.984
you see that 2/3 is twice 1/3.

00:02:09.984 --> 00:02:12.433
This right over here is twice as likely.

00:02:12.433 --> 00:02:15.233
You're twice as likely to
roll a number greater than

00:02:15.233 --> 00:02:18.000
or equal to three than
you are to roll a number

00:02:18.000 --> 00:02:19.389
less than or equal to two.

00:02:19.389 --> 00:02:20.559
You can even see right over here.

00:02:20.559 --> 00:02:22.366
You have twice as many possibilities

00:02:22.366 --> 00:02:26.000
of the six equally likely
ones, four versus two.

00:02:26.000 --> 00:02:28.340
Four versus two here.

00:02:28.340 --> 00:02:30.386
So you say, "Okay, I get it Sal."

00:02:30.386 --> 00:02:33.767
If the probability is a larger number,

00:02:33.767 --> 00:02:35.600
the event is more likely.

00:02:35.600 --> 00:02:37.466
It makes sense and in
this case, it's twice.

00:02:37.466 --> 00:02:40.600
The number is twice as large
so it's twice as likely.

00:02:40.600 --> 00:02:44.486
But what's the range of
possible probabilities?

00:02:44.486 --> 00:02:46.253
How low can a probability get

00:02:46.253 --> 00:02:49.100
and how high can a probability get?

00:02:49.100 --> 00:02:51.466
Let's think about the first question.

00:02:51.466 --> 00:02:55.650
How low can a probability go?

00:02:55.650 --> 00:02:59.200
How low, so what's the lowest probability

00:02:59.200 --> 00:03:01.858
that you can imagine for anything?

00:03:01.858 --> 00:03:06.771
Well, let's give ourselves a
little bit of an experiment.

00:03:06.771 --> 00:03:08.604
Let's ask ourselves the probability

00:03:08.604 --> 00:03:12.469
of rolling a seven.

00:03:13.560 --> 00:03:15.076
Well, once and pause the video

00:03:15.076 --> 00:03:17.155
and try to figure it out on your own.

00:03:17.155 --> 00:03:20.013
Well, there are six equally
likely possibilities.

00:03:20.013 --> 00:03:22.523
How many of them involve rolling a seven?

00:03:22.523 --> 00:03:23.925
Well, none of them.

00:03:23.925 --> 00:03:25.600
It's impossible to roll a seven.

00:03:25.600 --> 00:03:27.385
So none of the six.

00:03:27.385 --> 00:03:30.161
We could say this probability is zero.

00:03:30.161 --> 00:03:32.233
If you see a probability of zero,

00:03:32.233 --> 00:03:34.586
someone says the probability
of that thing happening

00:03:34.586 --> 00:03:36.934
is zero, that means it's impossible.

00:03:36.934 --> 00:03:40.500
That means in no world can that happen,

00:03:40.500 --> 00:03:42.254
if it's exactly zero.

00:03:42.254 --> 00:03:44.900
This right here, the probability is zero.

00:03:44.900 --> 00:03:48.104
That means it is impossible.

00:03:48.104 --> 00:03:51.718
It is impossible.

00:03:52.687 --> 00:03:54.900
Now how high can a probability get?

00:03:55.945 --> 00:03:59.325
How high

00:03:59.325 --> 00:04:02.107
can a probability get?

00:04:02.107 --> 00:04:03.061
Well, let's think about it.

00:04:03.061 --> 00:04:07.406
Let's say probability of rolling

00:04:07.406 --> 00:04:11.580
any number from one to six.

00:04:12.518 --> 00:04:17.162
Well, I have six equally

00:04:17.162 --> 00:04:20.176
likely possibilities
and any one of those six

00:04:20.176 --> 00:04:21.074
meets this constraint.

00:04:21.074 --> 00:04:23.511
I would have rolled a number, any number,

00:04:23.511 --> 00:04:25.961
from one to six, including one and six.

00:04:25.961 --> 00:04:29.100
So there are six equally
likely possibilities.

00:04:29.100 --> 00:04:31.562
So the probability is one.

00:04:31.562 --> 00:04:35.396
Someone says the probability
is zero, it's impossible.

00:04:35.396 --> 00:04:37.753
If someone says the probability is one,

00:04:37.753 --> 00:04:40.513
that means it's definitely
going to happen.

00:04:46.325 --> 00:04:48.433
It's definitely going to happen.

00:04:48.433 --> 00:04:52.000
So the maximum probability
for anything is one.

00:04:52.000 --> 00:04:53.558
The minimum probability is zero.

00:04:53.558 --> 00:04:55.373
You don't have negative probabilities

00:04:55.373 --> 00:04:58.225
and you don't have
probabilities greater than one.

00:04:58.225 --> 00:04:59.528
You might be thinking, "Wait, wait.

00:04:59.528 --> 00:05:01.700
"I've seen things that they look like

00:05:01.700 --> 00:05:03.600
"larger numbers than one."

00:05:03.600 --> 00:05:06.381
You're probably thinking of
seeing this as a percentage.

00:05:06.381 --> 00:05:09.066
One as a percentage,
you can also write this

00:05:09.066 --> 00:05:11.356
as 100%.

00:05:11.356 --> 00:05:14.746
This right over here as
a percentage is 100%.

00:05:14.746 --> 00:05:18.373
100% is the same thing as one.

00:05:18.373 --> 00:05:20.946
You can't have a probability at 110%.

00:05:20.946 --> 00:05:25.392
110% would be the same thing as 1.1.

00:05:25.392 --> 00:05:28.703
Now this is really interesting
because you'd often see

00:05:28.703 --> 00:05:31.085
someone say, "Hey, something
for sure is going to happen

00:05:31.085 --> 00:05:32.292
"or something is impossible."

00:05:32.292 --> 00:05:34.448
But even a lot of the things
that we think for sure

00:05:34.448 --> 00:05:36.727
are going to happen,
there's some probability

00:05:36.727 --> 00:05:38.848
or some chance that they don't happen.

00:05:38.848 --> 00:05:41.074
For example, you might hear someone say,

00:05:41.074 --> 00:05:43.566
"Well, what's the probability that the sun

00:05:43.566 --> 00:05:45.900
"will rise tomorrow?"

00:05:45.900 --> 00:05:48.833
Well, you might say it's
going to happen for sure.

00:05:48.833 --> 00:05:52.545
But you gotta remember some type of weird

00:05:52.545 --> 00:05:56.719
cosmological event might
occur, some kind of strange,

00:05:56.719 --> 00:06:01.233
huge planet-sized object
in space might come

00:06:01.233 --> 00:06:03.546
and knock the earth out of its rotation.

00:06:03.546 --> 00:06:05.492
Who knows what could happen?

00:06:05.492 --> 00:06:09.066
All these have a very low likelihood.

00:06:09.066 --> 00:06:12.066
Very, very, very, very,
very, very low likelihood.

00:06:12.066 --> 00:06:13.540
But it's hard to say it's exactly one.

00:06:13.540 --> 00:06:14.938
If I had said the probability

00:06:14.938 --> 00:06:19.400
that the sun will rise

00:06:19.400 --> 00:06:22.866
tomorrow,

00:06:22.866 --> 00:06:27.666
instead of saying one, I
would probably say it's 0.999.

00:06:27.666 --> 00:06:32.369
I would throw a lot of nines over here.

00:06:32.369 --> 00:06:35.423
I wouldn't say it's 0.9 repeating forever.

00:06:35.423 --> 00:06:38.379
Actually, there's an
interesting proof that 0.9

00:06:38.379 --> 00:06:40.981
repeating forever is actually
the same thing as one,

00:06:40.981 --> 00:06:42.700
which is a little counterintuitive.

00:06:42.700 --> 00:06:44.874
But I would say there's
a very high probability.

00:06:44.874 --> 00:06:46.866
But even if it's such a high probability,

00:06:46.866 --> 00:06:48.700
it's going to be close to one.

00:06:48.700 --> 00:06:50.338
But I won't say it's exactly the one

00:06:50.338 --> 00:06:52.925
because there could be some
kind of quasar that blasts us

00:06:52.925 --> 00:06:53.719
with gamma rays.

00:06:53.719 --> 00:06:56.163
Or who knows what might happen?

00:06:56.163 --> 00:06:58.446
But it's a very, very high probability.

00:06:58.446 --> 00:07:03.170
Same thing, the probability here,

00:07:03.170 --> 00:07:07.243
probability that my

00:07:07.243 --> 00:07:12.243
pet gopher

00:07:14.649 --> 00:07:17.030
could write the next great novel.

00:07:17.030 --> 00:07:21.353
Writes a novel.

00:07:21.353 --> 00:07:24.532
Actually not just a novel, a great novel.

00:07:24.532 --> 00:07:26.846
Just a novel wouldn't be
that impressive for a gopher.

00:07:26.846 --> 00:07:28.619
Let's say great novel.

00:07:28.619 --> 00:07:30.753
Well, once again, this
gopher sitting there

00:07:30.753 --> 00:07:32.056
typing at a keyboard.

00:07:32.056 --> 00:07:33.533
It would seem somewhat random

00:07:33.533 --> 00:07:38.074
but there is some probability
that it actually does it.

00:07:38.074 --> 00:07:39.453
There's some chance it does it.

00:07:39.453 --> 00:07:41.267
So I would put this at a very low

00:07:41.267 --> 00:07:42.900
and I wouldn't say it's exactly zero.

00:07:42.900 --> 00:07:45.992
If we had an infinite number
of gophers doing this forever,

00:07:45.992 --> 00:07:49.323
who knows, maybe one of them
might write that great novel.

00:07:49.323 --> 00:07:51.700
In fact, if we had an infinite
number doing it forever,

00:07:51.700 --> 00:07:53.800
eventually, a lot of people would say,

00:07:53.800 --> 00:07:55.133
at some point you would.

00:07:55.133 --> 00:07:57.753
But just one gopher
trying to write a novel,

00:07:57.753 --> 00:07:59.286
what's the probability
they write a great novel?

00:07:59.286 --> 00:08:01.236
I would say it's pretty close to zero.

00:08:01.236 --> 00:08:06.202
I'd throw a lot of zeros here,

00:08:06.202 --> 00:08:08.828
and at some point you might
have something like this.

00:08:08.828 --> 00:08:12.033
Once again, not absolutely impossible

00:08:12.033 --> 00:08:14.000
but pretty close to,

00:08:14.000 --> 00:08:18.063
pretty, pretty close to impossible.

00:08:18.063 --> 00:08:19.556
So big takeaways?

00:08:19.556 --> 00:08:22.100
Higher probability, more likely.

00:08:22.100 --> 00:08:24.333
The lowest probability
you can get to? Zero.

00:08:24.333 --> 00:08:26.200
Highest probability is one.

00:08:26.200 --> 00:08:27.966
If your probability is more,

00:08:27.966 --> 00:08:30.466
when you're talking about coin flipping.

00:08:30.466 --> 00:08:34.133
If you say the probability
of heads for a fair coin

00:08:34.133 --> 00:08:35.376
and you say, "Well, that's 1/2,"

00:08:35.376 --> 00:08:37.933
that means it's equally likely
to happen or not happen.

00:08:37.933 --> 00:08:40.900
Anything that has a larger
probability than 1/2,

00:08:40.900 --> 00:08:42.900
it's more likely to happen than not.

00:08:42.900 --> 00:08:45.402
Anything that has a
probability of less than 1/2,

00:08:45.402 --> 00:08:48.586
it's less likely to happen than not.


Experimental probability

WEBVTT
Kind: captions
Language: en

00:00:00.427 --> 00:00:01.870
- [Voiceover] There's a
lot of times, there's a lot

00:00:01.870 --> 00:00:04.157
of situations in which
we're studying something

00:00:04.157 --> 00:00:06.414
pretty straightforward and we can

00:00:06.414 --> 00:00:09.579
find an exact theoretical probability.

00:00:09.579 --> 00:00:11.135
So what am I talking about?

00:00:11.135 --> 00:00:12.326
Just let me write that down.

00:00:12.326 --> 00:00:15.947
Theoretical probabiity.

00:00:15.947 --> 00:00:17.955
Well, maybe the simplest example, or one

00:00:17.955 --> 00:00:21.344
of the simplest examples is
if you're flipping a coin.

00:00:21.344 --> 00:00:23.103
And let's say in theory you're flipping

00:00:23.103 --> 00:00:25.210
a completely fair coin and you're

00:00:25.210 --> 00:00:27.822
flipping it in a way
that is completely fair.

00:00:27.822 --> 00:00:31.226
Well, there you know
you have two outcomes.

00:00:31.226 --> 00:00:34.563
Either heads will be on top
or tails will be on top.

00:00:34.563 --> 00:00:36.428
So theoretically you say, "well, look,

00:00:36.428 --> 00:00:37.531
"if I want to figure out the probability

00:00:37.531 --> 00:00:41.025
"of getting a heads, in theory I have two

00:00:41.025 --> 00:00:43.813
"equally likely possibilities, and heads

00:00:43.813 --> 00:00:46.854
"is one of those two equally
likely possibilities."

00:00:46.854 --> 00:00:49.490
So you have a 1/2 probability.

00:00:49.490 --> 00:00:51.509
Once again, if in theory the coin

00:00:51.509 --> 00:00:54.939
is definitely fair, it's a
fair coin and it's flipped

00:00:54.939 --> 00:00:57.401
in a very fair way, then this is true.

00:00:57.401 --> 00:01:01.615
You have a 1/2 probability.

00:01:01.615 --> 00:01:04.076
We could also do that with rolling a die.

00:01:04.076 --> 00:01:07.023
A fair six-sided die is
going to have six possible

00:01:07.023 --> 00:01:10.621
outcomes: one, two,
three, four, five and six.

00:01:10.621 --> 00:01:14.267
And if you said "what is
the probability of getting

00:01:14.267 --> 00:01:17.712
"a result that is greater
than or equal to three?"

00:01:17.712 --> 00:01:20.382
Well, we have six equally
likely possibilities.

00:01:20.382 --> 00:01:21.663
You see them there.

00:01:21.663 --> 00:01:24.610
In theory, if they're all equally likely,

00:01:24.610 --> 00:01:27.423
four of these possibilities
meet our constraint

00:01:27.423 --> 00:01:29.876
of being greater than or equal to three.

00:01:29.876 --> 00:01:32.279
We have four out of the six of these

00:01:32.279 --> 00:01:34.241
possibilities meet our constraints.

00:01:34.241 --> 00:01:36.914
So we have a 2/3, 4/6 is the same thing

00:01:36.914 --> 00:01:40.374
as 2/3, probability of it happening.

00:01:40.374 --> 00:01:41.525
Now these are for simple things,

00:01:41.525 --> 00:01:45.334
like die or flipping a coin.

00:01:45.334 --> 00:01:47.272
And if you have fancy computers

00:01:47.272 --> 00:01:48.925
or spreadsheets you can even say "hey,

00:01:48.925 --> 00:01:50.377
"I'm gonna flip a coin a bunch of times

00:01:50.377 --> 00:01:52.687
"and do all the
combinatorics" and all that.

00:01:52.687 --> 00:01:54.138
But there are things that are even

00:01:54.138 --> 00:01:57.194
beyond what a computer can find

00:01:57.194 --> 00:01:59.243
the exact theoretical probability for.

00:01:59.243 --> 00:02:00.666
Let's say you are playing a game,

00:02:00.666 --> 00:02:03.074
say football, American football,

00:02:03.074 --> 00:02:04.454
and you wanted to figure
out the probability

00:02:04.454 --> 00:02:06.884
of scoring a certain number of points.

00:02:06.884 --> 00:02:09.332
Well that isn't very
simple because that's going

00:02:09.332 --> 00:02:12.251
to involve what human beings are doing.

00:02:12.251 --> 00:02:14.325
Minds are very unpredictable,

00:02:14.325 --> 00:02:15.950
how people will respond to things.

00:02:15.950 --> 00:02:18.983
The weather might get involved.

00:02:18.983 --> 00:02:20.334
Someone might fall sick.

00:02:20.334 --> 00:02:23.531
The ball might be wet,
or just how the ball

00:02:23.531 --> 00:02:25.945
might interact with some player's jersey.

00:02:25.945 --> 00:02:30.531
Who knows what might actually result

00:02:30.531 --> 00:02:34.303
in the score being one point this way,

00:02:34.303 --> 00:02:37.176
or seven points this way,
or seven points that way.

00:02:37.176 --> 00:02:39.832
So for situations like
that, it makes more sense

00:02:39.832 --> 00:02:42.897
to think more in terms of
experimental probability.

00:02:43.809 --> 00:02:46.467
In experimental probability,
we're really just trying

00:02:46.467 --> 00:02:48.721
to get an estimate of something happening,

00:02:48.721 --> 00:02:52.410
based on data and experience
that we've had in the past.

00:02:52.410 --> 00:02:57.286
For example, let's say you had data from

00:02:57.286 --> 00:03:02.224
your football team and it's
many games into the season.

00:03:02.224 --> 00:03:04.126
You've been tabulating
the number of points,

00:03:04.126 --> 00:03:06.424
you have a histogram
of the number of games

00:03:06.424 --> 00:03:08.153
that scored between zero and nine points.

00:03:08.153 --> 00:03:09.162
You had two games that scored

00:03:09.162 --> 00:03:10.660
between zero and nine points.

00:03:10.660 --> 00:03:15.660
Four games that scored
from ten to 19 points.

00:03:15.740 --> 00:03:18.594
You had five games that
went from 20 to 29 points.

00:03:18.594 --> 00:03:22.200
You had three games that
went from 30 to 39 points.

00:03:22.200 --> 00:03:25.475
Then you had two games that
go from 40 to 49 points.

00:03:25.475 --> 00:03:28.554
Now let's say for your
next game, let's see

00:03:28.554 --> 00:03:31.696
how many games you've had so far.

00:03:31.696 --> 00:03:34.549
The game so far is two plus four

00:03:34.549 --> 00:03:37.874
plus five plus three plus two,

00:03:37.874 --> 00:03:42.390
so this is six, plus five is 11.

00:03:42.390 --> 00:03:44.828
Eleven plus five is 16.

00:03:44.828 --> 00:03:47.216
So you've had 16 games so far this season

00:03:47.216 --> 00:03:49.898
and you're curious, for your 17th game,

00:03:53.439 --> 00:03:57.421
you want to figure out,
what is the probability

00:04:08.717 --> 00:04:12.828
your points are greater
than or equal to 30?

00:04:12.828 --> 00:04:13.834
The probability your
points are greater than

00:04:13.834 --> 00:04:16.710
or equal to 30 for game 17.

00:04:16.710 --> 00:04:18.602
Once again, this is very hard to find

00:04:18.602 --> 00:04:20.680
the exact theoretical probability.

00:04:20.680 --> 00:04:23.530
You don't know exactly, you
can't predict the future.

00:04:23.530 --> 00:04:25.795
You don't know who's gonna show up sick,

00:04:25.795 --> 00:04:27.837
how humans are going to
interact with each other.

00:04:27.837 --> 00:04:29.974
Maybe someone screams
something in the stand

00:04:29.974 --> 00:04:31.504
that just phases the quarterback

00:04:31.504 --> 00:04:33.443
in exactly the right or the wrong way.

00:04:33.443 --> 00:04:34.254
You don't know.

00:04:34.254 --> 00:04:36.004
This is an incredibly, incredibly complex

00:04:36.004 --> 00:04:37.555
system, what might happen over

00:04:37.555 --> 00:04:39.863
the course of an entire football game.

00:04:39.863 --> 00:04:41.661
But you can estimate what'll happen based

00:04:41.661 --> 00:04:43.594
on what you've seen in
your past experience.

00:04:43.594 --> 00:04:45.243
It depends on the defense of the team

00:04:45.243 --> 00:04:46.402
you're facing and all that.

00:04:46.402 --> 00:04:50.094
So it's not going to be super exact,

00:04:50.094 --> 00:04:52.361
but you could estimate,
based on experiments,

00:04:52.361 --> 00:04:54.204
based on what you've seen in the past.

00:04:54.204 --> 00:04:56.726
Here, the experimental probability,

00:04:56.726 --> 00:04:59.773
and I would say the estimate,

00:04:59.773 --> 00:05:03.407
because you shouldn't walk away saying,

00:05:03.407 --> 00:05:05.032
"okay, we absolutely know for sure

00:05:05.032 --> 00:05:07.324
"that if we conducted this next game

00:05:07.324 --> 00:05:08.998
"experiment n times that it's definitely

00:05:08.998 --> 00:05:09.725
"gonna turn out the same."

00:05:09.725 --> 00:05:11.706
Because this might be the toughest

00:05:11.706 --> 00:05:12.754
defense that you play all year, this might

00:05:12.754 --> 00:05:15.192
be the easiest defense
that you play all year.

00:05:15.192 --> 00:05:16.538
But if you look at what's happened

00:05:16.538 --> 00:05:20.431
in the past, out of the 16 games so far,

00:05:20.431 --> 00:05:23.949
there have been three games, these three

00:05:23.949 --> 00:05:27.448
plus these two games, where you scored

00:05:27.448 --> 00:05:30.190
greater than or equal to 30 points.

00:05:30.190 --> 00:05:33.255
So five out of the 16 situations,

00:05:33.255 --> 00:05:35.068
you've scored more than that.

00:05:35.068 --> 00:05:37.767
An estimate of your probability, you could

00:05:37.767 --> 00:05:40.286
view this as maybe your
experimental probability,

00:05:40.286 --> 00:05:42.603
of scoring more than
30 points based on past

00:05:42.603 --> 00:05:47.603
experience, based on
past experience, is five,

00:05:48.186 --> 00:05:51.640
five out of the 16 games
you've done this in the past.

00:05:51.640 --> 00:05:53.673
So you'd say it's 5/16.

00:05:53.673 --> 00:05:55.159
Now I want to really have you

00:05:55.159 --> 00:05:56.657
take this with a grain of salt.

00:05:56.657 --> 00:06:00.287
You should not say "okay, I know for sure

00:06:00.287 --> 00:06:03.470
"there's a 5/16 probability
of us winning this game."

00:06:03.470 --> 00:06:04.813
Because you only have some data points,

00:06:04.813 --> 00:06:06.591
every team you play is
going to be different,

00:06:06.591 --> 00:06:08.006
it's going to be different
weather conditions,

00:06:08.006 --> 00:06:10.445
people are going to be in
different moods, etc., etc.

00:06:10.445 --> 00:06:13.305
This is really just an estimate.

00:06:13.305 --> 00:06:14.476
I feel a little bit of reservations

00:06:14.476 --> 00:06:16.056
even calling it a probability.

00:06:16.056 --> 00:06:18.158
I would just say that this has been true

00:06:18.158 --> 00:06:20.328
of five out of 16 games in the past.

00:06:20.328 --> 00:06:24.629
So it's an indicator of what might be.

00:06:24.629 --> 00:06:27.424
You might say, "okay, based
on experience, it's more

00:06:27.424 --> 00:06:31.382
"likely than not that we don't
score more than 30 points."

00:06:31.382 --> 00:06:34.446
But it's really just based
on experiential data,

00:06:34.446 --> 00:06:35.580
what's happened in the season.

00:06:35.580 --> 00:06:36.666
Even the makeup of your football

00:06:36.666 --> 00:06:37.651
team might have changed.

00:06:37.651 --> 00:06:39.108
You might have gotten a different coach,

00:06:39.108 --> 00:06:40.843
you might have learned to train better.

00:06:40.843 --> 00:06:42.631
Who knows, one of your team members

00:06:42.631 --> 00:06:44.407
might have grown by three inches.

00:06:44.407 --> 00:06:45.175
All of these things.

00:06:45.175 --> 00:06:47.892
So all of this has to be
taken with a grain of salt.

00:06:47.892 --> 00:06:49.575
But this is one way of thinking about it.

00:06:49.575 --> 00:06:53.341
At least having a sense
of what may happen.


Thinking about shapes of distributions

WEBVTT
Kind: captions
Language: en

00:00:00.290 --> 00:00:01.417
- [Voiceover] So what I
want to talk about now

00:00:01.417 --> 00:00:04.155
are shapes of distributions
and different words

00:00:04.155 --> 00:00:06.431
we might use to describe those shapes.

00:00:06.431 --> 00:00:08.149
So right over here, let's see,

00:00:08.149 --> 00:00:09.658
we're talking about Matt's Cafe,

00:00:09.658 --> 00:00:11.237
and we have different age buckets,

00:00:11.237 --> 00:00:12.874
so this is a histogram here.

00:00:12.874 --> 00:00:14.255
In each bucket, it tells us the number

00:00:14.255 --> 00:00:16.856
of guests that are in that age bucket.

00:00:16.856 --> 00:00:18.632
So we don't have any guests that are

00:00:18.632 --> 00:00:20.861
under the age of 20, we
have a reasonable number

00:00:20.861 --> 00:00:23.009
between 20 and 30, we have a lot of guests

00:00:23.009 --> 00:00:25.645
at 30, in that bucket between 30 and 40,

00:00:25.645 --> 00:00:27.479
reasonable number between 40 and 50,

00:00:27.479 --> 00:00:28.235
and then as we get older,

00:00:28.235 --> 00:00:30.217
we have fewer and fewer guests.

00:00:30.300 --> 00:00:31.821
So just when you look
at something like this,

00:00:31.821 --> 00:00:32.680
a distribution like this,

00:00:32.680 --> 00:00:33.981
something might pop out at you.

00:00:33.981 --> 00:00:35.525
It kind of looks like
if you were to imagine

00:00:35.525 --> 00:00:37.486
this were an armadillo, this would be

00:00:37.486 --> 00:00:39.065
the body of the armadillo,

00:00:39.065 --> 00:00:40.122
and then what we see to the right

00:00:40.122 --> 00:00:42.571
kind of looks like the
tail of the armadillo.

00:00:42.571 --> 00:00:44.679
We actually use those types of words

00:00:44.679 --> 00:00:46.467
to describe distributions.

00:00:46.467 --> 00:00:47.940
So this distribution right over here,

00:00:47.940 --> 00:00:49.787
it looks like it has a tail to the right.

00:00:49.787 --> 00:00:51.064
It doesn't have a tail to the left.

00:00:51.064 --> 00:00:53.862
In fact, we have no one
under the age of 20.

00:00:53.862 --> 00:00:55.684
But here when we have a few people

00:00:55.684 --> 00:00:58.451
between 60 and 70, even
fewer between 70 and 80,

00:00:58.451 --> 00:00:59.590
even fewer between 80 and 90,

00:00:59.590 --> 00:01:00.891
and you know, if it just kind of keeps

00:01:00.891 --> 00:01:02.794
going like this, this is a tail

00:01:02.794 --> 00:01:03.932
and it's on the right side,

00:01:03.932 --> 00:01:05.779
it's a right-tailed distribution.

00:01:05.779 --> 00:01:07.468
So I'd call this
distribution right-tailed.

00:01:07.468 --> 00:01:09.441
I'm using Khan Academy exercises

00:01:09.441 --> 00:01:11.531
because it's a good way
to see a lot of examples,

00:01:11.531 --> 00:01:13.690
and frankly, you should too because it'll

00:01:13.690 --> 00:01:15.409
help you test your knowledge.

00:01:15.409 --> 00:01:16.302
But it's not left-tailed.

00:01:16.302 --> 00:01:18.567
Left-tailed we would see
a tail going like that.

00:01:18.567 --> 00:01:20.748
Frankly, if you're
left-tailed and right-tailed,

00:01:20.748 --> 00:01:22.700
you're likely to be
approximately symmetrical.

00:01:22.700 --> 00:01:25.173
Remember symmetry, you
define a line of symmetry,

00:01:25.173 --> 00:01:28.817
and one type of symmetry
is one where both sides

00:01:28.817 --> 00:01:30.558
of that line of symmetry are mirror images

00:01:30.558 --> 00:01:31.347
of each other.

00:01:31.347 --> 00:01:32.950
You could fold over the line of symmetry,

00:01:32.950 --> 00:01:34.847
and they'll roughly meet.

00:01:34.847 --> 00:01:36.508
This one does not meet
that because if you were

00:01:36.508 --> 00:01:38.784
to say, hey, maybe there's
a line of symmetry here

00:01:38.784 --> 00:01:40.234
and you tried to fold this over,

00:01:40.234 --> 00:01:41.187
it wouldn't match up,

00:01:41.187 --> 00:01:43.149
the two sides would not match up.

00:01:43.149 --> 00:01:45.946
So I feel good saying
that it is right-tailed.

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So let's see.

00:01:47.090 --> 00:01:50.087
Retirement of age of each guest.

00:01:50.087 --> 00:01:52.014
Well yeah, these names aren't that great,

00:01:52.014 --> 00:01:53.313
but let's actually see
what they're saying.

00:01:53.313 --> 00:01:55.554
They're saying by age, they're telling us

00:01:55.554 --> 00:01:57.722
the number of guests.

00:01:57.722 --> 00:01:59.739
So this is the number of guests

00:01:59.739 --> 00:02:01.171
at Logan Assisted Living.

00:02:01.171 --> 00:02:03.642
So we have a lot of guests that are

00:02:03.642 --> 00:02:05.279
between 60 and 70 years old,

00:02:05.279 --> 00:02:07.079
or reasonable that are between 50 and 60,

00:02:07.079 --> 00:02:09.309
or 70 or 80, and this distribution

00:02:09.309 --> 00:02:11.501
actually looks pretty symmetrical.

00:02:11.501 --> 00:02:13.220
If I were to draw a line of symmetry

00:02:13.220 --> 00:02:16.285
right down here, right
at around an age of,

00:02:16.285 --> 00:02:17.829
you know, the line would be right at

00:02:17.829 --> 00:02:19.262
an age of 65, I guess you could say

00:02:19.262 --> 00:02:22.038
all this is a bucket for ages 60 to 70,

00:02:22.038 --> 00:02:23.175
then you could flip it over

00:02:23.175 --> 00:02:24.668
and it looks pretty symmetrical.

00:02:24.668 --> 00:02:27.338
Not exactly, this bucket
doesn't quite match

00:02:27.338 --> 00:02:28.940
up to this one, but it's pretty close.

00:02:28.940 --> 00:02:30.635
These roughly match each other.

00:02:30.844 --> 00:02:32.807
These roughly match each other.

00:02:32.807 --> 00:02:33.991
So I feel good about saying

00:02:33.991 --> 00:02:36.264
it is approximately symmetrical.

00:02:36.731 --> 00:02:38.205
Now just to know what
these other words mean,

00:02:38.205 --> 00:02:41.885
skewed to the left, or
skewed to the right.

00:02:41.885 --> 00:02:43.893
These actually have fairly
technical definitions

00:02:43.893 --> 00:02:46.297
when you get further in statistics,

00:02:46.297 --> 00:02:51.091
but a, I guess, easier to
process version of them

00:02:51.091 --> 00:02:56.091
are when you're left tailed,

00:02:56.594 --> 00:02:58.847
you also tend to be skewed to the left,

00:02:58.847 --> 00:03:00.112
and when you are right-tailed,

00:03:00.112 --> 00:03:02.271
you tend to be skewed to the right.

00:03:02.271 --> 00:03:04.790
Another way to think
about skewed to the left

00:03:04.790 --> 00:03:07.113
is that your mean is to the left

00:03:07.113 --> 00:03:09.132
of your median and mode.

00:03:09.132 --> 00:03:10.292
That might not make any sense to you.

00:03:10.292 --> 00:03:11.483
You might just want to off of the tail.

00:03:11.483 --> 00:03:13.967
If you're left-tailed,
you're probably left skewed.

00:03:13.967 --> 00:03:16.493
If you're right-tailed,
you're probably right skewed.

00:03:16.493 --> 00:03:17.269
So let's keep going.

00:03:18.514 --> 00:03:19.866
Let's see another example.

00:03:19.866 --> 00:03:20.948
So this interesting.

00:03:21.687 --> 00:03:23.506
We're not given a histogram here.

00:03:23.506 --> 00:03:24.680
We're not given a bar graph.

00:03:24.680 --> 00:03:26.479
We're given a box and whiskers plot,

00:03:26.479 --> 00:03:27.524
which is really just telling us

00:03:27.524 --> 00:03:28.848
the different quartiles.

00:03:28.848 --> 00:03:30.647
So just to remind ourselves,

00:03:30.647 --> 00:03:33.480
this tells us the minimum of our data set,

00:03:33.480 --> 00:03:34.884
the bottom of our range,

00:03:34.884 --> 00:03:36.661
so the minimum value in our data set.

00:03:36.661 --> 00:03:39.621
We have at least one 11,

00:03:39.621 --> 00:03:41.281
and then the maximum
value of our data set,

00:03:41.281 --> 00:03:43.986
we have at least one 25.

00:03:43.986 --> 00:03:46.006
Now this line right
over here is the median.

00:03:46.006 --> 00:03:48.746
The middle number is 21.

00:03:48.746 --> 00:03:51.903
Then the box defines the middle 50%

00:03:51.903 --> 00:03:53.679
of our numbers.

00:03:53.679 --> 00:03:58.108
So it's kind of the meat
of our distribution.

00:03:58.108 --> 00:03:59.654
So if we were to try to visualize

00:03:59.654 --> 00:04:01.586
what this would look like
as maybe a histogram,

00:04:01.586 --> 00:04:02.991
and we don't know for
sure because we might

00:04:02.991 --> 00:04:05.218
have a whole bunch of 11s,

00:04:05.218 --> 00:04:06.945
not so much that it skews this,

00:04:06.945 --> 00:04:08.605
but we could have more than one.

00:04:08.605 --> 00:04:10.764
But a distribution that this could match

00:04:10.764 --> 00:04:12.313
up with is something that looks like

00:04:12.313 --> 00:04:14.249
having a tail down here,

00:04:14.249 --> 00:04:15.524
and then you kind of bump up here.

00:04:15.524 --> 00:04:16.889
This is the meat of the distribution.

00:04:16.889 --> 00:04:18.967
It kind of looks something like that,

00:04:18.967 --> 00:04:20.499
and I can't draw because I'm doing this

00:04:20.499 --> 00:04:22.666
on the exercises right now.

00:04:22.666 --> 00:04:24.125
But for something like that,

00:04:24.125 --> 00:04:25.611
well something like that would have a tail

00:04:25.611 --> 00:04:28.430
to the left, would have
a tail to the left.

00:04:29.475 --> 00:04:33.221
It's range goes fairly low to the left,

00:04:33.221 --> 00:04:34.858
but it might not have
a lot of value there.

00:04:34.858 --> 00:04:36.402
If I had more values on the left side,

00:04:36.402 --> 00:04:38.201
this box would have been shifted over

00:04:38.201 --> 00:04:42.056
because a larger
percentage would have been

00:04:42.056 --> 00:04:44.122
on the left, so to speak.

00:04:44.122 --> 00:04:46.258
So this one, I feel
pretty good about saying

00:04:46.258 --> 00:04:48.313
this is skewed to the left.

00:04:48.313 --> 00:04:49.579
It's definitely not symmetrical.

00:04:49.579 --> 00:04:52.459
If it was symmetrical, the median would be

00:04:52.459 --> 00:04:53.947
pretty close to the center,

00:04:53.947 --> 00:04:56.513
the box would be pretty centered.

00:04:56.513 --> 00:04:57.707
It's not skewed to the right.

00:04:57.707 --> 00:04:58.707
If it was skewed to the right,

00:04:58.707 --> 00:05:00.227
you would have a tail to the right,

00:05:00.227 --> 00:05:01.934
you would have, this whisker would likely

00:05:01.934 --> 00:05:04.030
be much, much, much longer.

00:05:05.110 --> 00:05:06.085
And we're done.