Addition rule for probability

WEBVTT
Kind: captions
Language: en

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Let's say I have a bag.

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And in that bag--
I'm going to put

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some green cubes in that bag.

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And in particular, I'm going
to put eight green cubes.

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I'm also going to put
some spheres in that bag.

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Let's say I'm going
to put nine spheres.

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And these are the green spheres.

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I'm also going to put some
yellow cubes in that bag.

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I'm going to put five of those.

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And I'm also going to put some
yellow spheres in this bag.

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And let's say I
put seven of those.

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I'm going to stick
them all in this bag.

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And then I'm going
to shake that bag.

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And I'm going to pour it out.

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And I'm going to look
at the first object that

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falls out of that bag.

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And what I want to think
about in this video

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is what are the
probabilities of getting

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different types of objects?

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So for example, what
is the probability

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of getting a cube of any color?

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What is the probability
of getting a cube?

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Well, to think about that
we should think about what--

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or this is one way
to think about it--

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what are all of the equally
likely possibilities that

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might pop out of the bag?

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Well, we have 8 plus 9 is 17.

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17 plus 5 is 22.

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22 plus 7 is 29.

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So we have 29 objects.

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There are 29 objects in the bag.

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Did I do that right?

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This is 14, yup 29 objects.

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So let's draw all of
the possible objects.

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I'll represent it as this
big area right over here.

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So these are all the
possible objects.

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There are 29 possible objects.

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So there's 29
equal possibilities

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for the outcome of my experiment
of seeing what pops out

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of the bag, assuming
that it's equally

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likely for a cube or a
sphere to pop out first.

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And how many of them meet our
constraint of being a cube?

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Well, I have eight green cubes,
and I have five yellow cubes.

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So there are a
total of 13 cubes.

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So let me draw
that set of cubes.

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So there's 13 cubes.

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We could draw it like
this-- there are 13 cubes.

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This right here is the
set of cubes, this area.

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And I'm not drawing it exact.

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I'm approximating.

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It represents the
set of all the cubes.

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So the probability
of getting a cube

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is the number of events
that meet our criteria.

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So there's 13
possible cubes that

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have an equally likely
chance of popping out,

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over all of the possible equally
likely events, which are 29.

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That includes the
cubes and the spheres.

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Now let's ask a
different question.

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What is the probability of
getting a yellow object, either

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a cube or a sphere?

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So once again, how many things
meet our conditions here?

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Well, we have 5 plus 7.

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There's 12 yellow
objects in the bag.

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So we have 29 equally
likely possibilities.

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I'll do it in that same color.

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We have 29 equally
likely possibilities.

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And of those, 12
meet our criteria.

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So let me draw 12
right over here.

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I'll do my best attempt.

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So let's say it looks
something like-- so

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the set of yellow objects.

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There are 12 objects
that are yellow.

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So the 12 that
meet our conditions

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are 12, over all the
possibilities-- 29.

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So the probability
of getting a cube--

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13 29ths, probability of
getting a yellow-- 12 29ths.

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Now let's ask something a
little bit more interesting.

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What is the probability
of getting a yellow cube?

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So I'll put it in yellow.

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So we care about the color, now.

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So this thing is yellow.

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What is the probability of-- or
as my son would say, "lello."

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What is the probability
of getting a yellow cube?

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Well, there's 29 equally
likely possibilities.

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And of those 29 equally likely
possibilities, 5 of those

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are yellow cubes, or
"lello" cubes, five of them.

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So the probability is 5 29ths.

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And where would we see
that on this Venn diagram

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that I've drawn?

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This Venn diagram is
just a way to visualize

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the different probabilities.

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And they become
interesting when you

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start thinking about
where sets overlap,

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or even where they
don't overlap.

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So here we are
thinking about things

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that are members
of the set yellow.

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So they're in this set,
and they are cubes.

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So this area right
over here-- that's

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the overlap of these two sets.

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So this area right
over here-- this

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represents things that
are both yellow and cubes,

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because they are
inside both circles.

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So this right over here-- let
me rewrite it right over here.

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So there's five objects that
are both yellow and cubes.

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Now let's ask-- and this is
probably the most interesting

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thing to ask-- what is
the probability of getting

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something that is yellow or or
a cube, a cube of any color?

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The probability of getting
something that is yellow

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or a cube of any
color-- well, we still

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know that the denominator
here is going to be 29.

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These are all of the
equally likely possibilities

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that might jump out of the bag.

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But what are the possibilities
that meet our conditions?

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Well, one way to think about
it is, well, the probability--

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there's 12 things that would
meet the yellow condition.

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So that would be this entire
circle right over here-- 12

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things that meet the
yellow condition.

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So this right over here is 12.

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This is the number of yellow.

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That is 12.

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And then to that, we can't
just add the number of cubes,

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because if we add
the number of cubes,

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we've already counted these 5.

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These 5 are counted
as part of this 12.

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One way to think
about it is there

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are 7 yellow objects
that are not cubes.

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Those are the spheres.

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There are 5 yellow
objects that are cubes.

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And then there are 8
cubes that are not yellow.

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That's one way to think about.

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So when we counted this
12-- the number of yellow--

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we counted all of this.

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So we can't just add
the number of cubes

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to it, because then we would
count this middle part again.

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So then we have to
essentially count

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cubes, the number of
cubes, which is 13.

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So the number of
cubes, and we'll

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have to subtract out this
middle section right over here.

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Let me do this.

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So subtract out the middle
section right over here.

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So minus 5.

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So this is the number
of yellow cubes.

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It feels weird to write
the word yellow in green.

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The number of yellow cubes-- or
another way to think about it--

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and you could just do
this math right here.

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12 plus 13 minus 5 is 20.

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Did I do that right?

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12 minus, yup, it's 20.

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So that's one way.

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You just get this is
equal to 20 over 29.

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But the more interesting
thing than even the answer

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of the probability
of getting that,

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is expressing this in terms
of the other probabilities

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that we figured out
earlier in the video.

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So let's think about
this a little bit.

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We can rewrite this
fraction right over here.

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We can rewrite this as 12 over
29 plus 13 over 29 minus 5

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over 29.

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And this was the
number of yellow

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over the total possibilities.

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So this right over here
was the probability

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of getting a yellow.

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This right over
here was the number

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of cubes over the
total possibilities.

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So this is plus the
probability of getting a cube.

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And this right over
here is the number

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of yellow cubes over
the total possibilities.

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So this right over here
was minus the probability

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of yellow, and a cube.

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I'm not going to
write it that way.

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Minus the probability
of yellow-- I'll

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write yellow in yellow--
yellow and a cube.

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And so what we've
just done here--

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and you could play
with the numbers.

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The numbers I just used
as an example right

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here to make things a
little bit concrete.

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But you can see this is
a generalizable thing.

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If we have the probability
of one condition

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or another condition-- so let me
rewrite it-- the probability--

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and I'll just write it a
little bit more generally here.

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This gives us an
interesting idea.

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The probability of getting
one condition of an object

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being a member of set
a, or a member of set b

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is equal to the probability
that it is a member of set a,

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plus the probability that
is a member of set b,

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minus the probability
that is a member of both.

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And this is a really
useful result.

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I think sometimes it's
called the addition

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rule of probability.

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But I want to show you that
it's a completely common-sense

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thing.

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The reason why you can't just
add these two probabilities

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is because they might
have some overlap.

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There's a probability
of getting both.

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And if you just
added both of these,

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you would be double
counting that overlap,

00:09:13.590 --> 00:09:16.070
which we've already seen
earlier in this video.

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So you have to subtract
one version of the overlap

00:09:18.570 --> 00:09:21.640
out so you are not
double counting it.

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I'll throw another
one other idea out.

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Sometimes, you
have possibilities

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that have no overlap.

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So let's say this is the
set of all possibilities.

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And let's say this is the
set that meets condition a

00:09:37.970 --> 00:09:40.620
and let me do this
in a different color.

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And let's say that this is the
set that meets condition b.

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So in this situation,
there is no overlap.

00:09:45.730 --> 00:09:48.580
There's no way-- nothing is a
member of both sets, a and b.

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So in this situation, the
probability of a and b is 0.

00:09:52.490 --> 00:09:54.200
There is no overlap.

00:09:54.200 --> 00:09:58.600
And these type of conditions,
or these two events,

00:09:58.600 --> 00:10:00.395
are called mutually exclusive.

00:10:05.310 --> 00:10:06.810
So if events are
mutually exclusive,

00:10:06.810 --> 00:10:11.420
that means that they both
cannot happen at the same time.

00:10:11.420 --> 00:10:15.800
There's no event that meets
both of these conditions.

00:10:15.800 --> 00:10:18.810
And if things are
mutually exclusive,

00:10:18.810 --> 00:10:21.410
then you can say the
probability of a or b

00:10:21.410 --> 00:10:24.950
is the probability of a plus
b, because this thing is 0.

00:10:24.950 --> 00:10:27.170
But if things are not
mutually exclusive,

00:10:27.170 --> 00:10:29.240
you would have to
subtract out the overlap.

00:10:29.240 --> 00:10:31.360
And probably the
best way to think

00:10:31.360 --> 00:10:33.320
about it is to
just always realize

00:10:33.320 --> 00:10:35.400
that you have to
subtract out the overlap.

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And obviously if something
is mutually exclusive,

00:10:38.170 --> 00:10:41.855
the probability of getting
a and b is going to be 0.


Probability with playing cards and Venn diagrams

WEBVTT
Kind: captions
Language: en

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Let's do a little bit of
probability with playing cards.

00:00:02.970 --> 00:00:04.511
And for the sake of
this video, we're

00:00:04.511 --> 00:00:06.864
going to assume that our
deck has no jokers in it.

00:00:06.864 --> 00:00:08.780
You could do the same
problems with the joker,

00:00:08.780 --> 00:00:11.560
you'll just get slightly
different numbers.

00:00:11.560 --> 00:00:13.390
So with that out of
the way, let's first

00:00:13.390 --> 00:00:15.800
just think about
how many cards we

00:00:15.800 --> 00:00:18.210
have in a standard playing deck.

00:00:18.210 --> 00:00:23.040
So you have four
suits, and the suits

00:00:23.040 --> 00:00:26.960
are the spades, the diamonds,
the clubs, and the hearts.

00:00:26.960 --> 00:00:29.770
You have four suits and
then in each of those suits

00:00:29.770 --> 00:00:31.950
you have 13 different
types of cards--

00:00:31.950 --> 00:00:33.640
and sometimes it's
called the rank.

00:00:43.970 --> 00:00:47.920
You have the ace, then you have
the two, the three, the four,

00:00:47.920 --> 00:00:52.210
the five, the six,
seven, eight, nine, ten,

00:00:52.210 --> 00:00:55.620
and then you have the Jack,
the King, and the Queen.

00:00:55.620 --> 00:00:57.590
And that is 13 cards.

00:00:57.590 --> 00:01:00.890
So for each suit
you can have any

00:01:00.890 --> 00:01:03.480
of these-- you can
have any of the suits.

00:01:03.480 --> 00:01:05.910
So you could have a Jack of
diamonds, a Jack of clubs,

00:01:05.910 --> 00:01:09.220
a Jack of spades,
or a Jack of hearts.

00:01:09.220 --> 00:01:10.970
So if you just multiply
these two things--

00:01:10.970 --> 00:01:13.532
you could take a deck of playing
cards, take out the jokers

00:01:13.532 --> 00:01:15.240
and count them-- but
if you just multiply

00:01:15.240 --> 00:01:18.740
this you have four suits, each
of those suits have 13 types.

00:01:18.740 --> 00:01:21.120
So you're going to
have 4 times 13 cards,

00:01:21.120 --> 00:01:24.207
or you're going to have 52 cards
in a standard playing deck.

00:01:24.207 --> 00:01:25.790
Another way you could
have said, look,

00:01:25.790 --> 00:01:28.270
there's 13 of these
ranks, or types,

00:01:28.270 --> 00:01:31.120
and each of those come in four
different suits-- 13 times 4.

00:01:31.120 --> 00:01:33.237
Once again, you would
have gotten 52 cards.

00:01:33.237 --> 00:01:34.820
Now, with that of
the way, let's think

00:01:34.820 --> 00:01:37.156
about the probabilities
of different events.

00:01:37.156 --> 00:01:38.530
So let's say I
shuffle that deck.

00:01:38.530 --> 00:01:40.610
I shuffle it really,
really well and then

00:01:40.610 --> 00:01:43.240
I randomly pick a
card from that deck.

00:01:43.240 --> 00:01:46.400
And I want to think about
what is the probability that I

00:01:46.400 --> 00:01:50.220
pick a Jack.

00:01:50.220 --> 00:01:53.450
Well, how many equally
likely events are there?

00:01:53.450 --> 00:01:56.540
Well, I could pick any
one of those 52 cards.

00:01:56.540 --> 00:02:00.350
So there's 52 possibilities
for when I pick that card.

00:02:00.350 --> 00:02:04.130
And how many of those 52
possibilities are Jacks?

00:02:04.130 --> 00:02:07.480
Well you have the Jack of
spades, the Jack of diamonds,

00:02:07.480 --> 00:02:09.880
the Jack of clubs, and
the Jack of hearts.

00:02:09.880 --> 00:02:14.170
There's four Jacks in that deck.

00:02:14.170 --> 00:02:18.090
So it is 4 over 52-- these
are both divisible by 4-- 4

00:02:18.090 --> 00:02:23.060
divided by 4 is 1, 52
divided by 4 is 13.

00:02:23.060 --> 00:02:27.390
Now, let's think
about the probability.

00:02:27.390 --> 00:02:28.976
So I'll start over.

00:02:28.976 --> 00:02:30.600
I'm going to put that
Jack back and I'm

00:02:30.600 --> 00:02:31.850
going to reshuffle the deck.

00:02:31.850 --> 00:02:34.020
So once again, I
still have 52 cards.

00:02:34.020 --> 00:02:37.290
So what's the probability
that I get a hearts?

00:02:37.290 --> 00:02:39.290
What's the probability
that I just randomly pick

00:02:39.290 --> 00:02:43.720
a card from a shuffled
deck and it is a heart?

00:02:43.720 --> 00:02:46.580
Well, once again,
there's 52 possible cards

00:02:46.580 --> 00:02:47.480
I could pick from.

00:02:47.480 --> 00:02:51.720
52 possible, equally likely
events that we're dealing with.

00:02:51.720 --> 00:02:55.010
And how many of those
have our hearts?

00:02:55.010 --> 00:02:57.770
Well, essentially 13
of them are hearts.

00:02:57.770 --> 00:03:00.180
For each of those suits
you have 13 types.

00:03:00.180 --> 00:03:01.892
So there are 13
hearts in that deck.

00:03:01.892 --> 00:03:03.350
There are 13 diamonds
in that deck.

00:03:03.350 --> 00:03:04.970
There are 13 spades
in that deck.

00:03:04.970 --> 00:03:07.490
There are 13 clubs in that deck.

00:03:07.490 --> 00:03:12.570
So 13 of the 52 would result
in hearts, and both of these

00:03:12.570 --> 00:03:14.270
are divisible by 13.

00:03:14.270 --> 00:03:16.670
This is the same thing as 1/4.

00:03:16.670 --> 00:03:19.040
One in four times
I will pick it out,

00:03:19.040 --> 00:03:21.690
or I have a one in four
probability of getting a hearts

00:03:21.690 --> 00:03:24.619
when I randomly pick a card
from that shuffled deck.

00:03:24.619 --> 00:03:27.160
Now, let's do something that's
a little bit more interesting,

00:03:27.160 --> 00:03:29.250
or maybe it's a little obvious.

00:03:29.250 --> 00:03:32.130
What's the probability
that I pick something

00:03:32.130 --> 00:03:38.540
that is a Jack-- I'll just
write J-- and it is a hearts?

00:03:42.120 --> 00:03:44.230
Well, if you are reasonably
familiar with cards

00:03:44.230 --> 00:03:45.605
you'll know that
there's actually

00:03:45.605 --> 00:03:47.900
only one card that is
both a Jack and a heart.

00:03:47.900 --> 00:03:49.520
It is literally
the Jack of hearts.

00:03:49.520 --> 00:03:50.530
So we're saying, what
is the probability

00:03:50.530 --> 00:03:52.870
that we pick the exact
card, the Jack of hearts?

00:03:52.870 --> 00:03:56.200
Well, there's only
one event, one card,

00:03:56.200 --> 00:04:00.660
that meets this criteria
right over here,

00:04:00.660 --> 00:04:02.630
and there's 52 possible cards.

00:04:02.630 --> 00:04:04.630
So there's a one
in 52 chance that I

00:04:04.630 --> 00:04:07.650
pick the Jack of hearts--
something that is both a Jack

00:04:07.650 --> 00:04:09.710
and it's a heart.

00:04:09.710 --> 00:04:12.640
Now, let's do something a
little bit more interesting.

00:04:12.640 --> 00:04:14.434
What is the
probability-- you might

00:04:14.434 --> 00:04:16.600
want to pause this and think
about this a little bit

00:04:16.600 --> 00:04:17.808
before I give you the answer.

00:04:17.808 --> 00:04:20.019
What is the probability
of-- so I once again, I

00:04:20.019 --> 00:04:22.340
have a deck of 52
cards, I shuffled it,

00:04:22.340 --> 00:04:25.350
randomly pick a card from that
deck-- what is the probability

00:04:25.350 --> 00:04:31.420
that that card that I pick from
that deck is a Jack or a heart?

00:04:31.420 --> 00:04:33.050
So it could be the
Jack of hearts,

00:04:33.050 --> 00:04:35.414
or it could be the
Jack of diamonds,

00:04:35.414 --> 00:04:36.830
or it could be the
Jack of spades,

00:04:36.830 --> 00:04:38.450
or it could be the
Queen of hearts,

00:04:38.450 --> 00:04:39.872
or it could be
the two of hearts.

00:04:39.872 --> 00:04:41.330
So what is the
probability of this?

00:04:41.330 --> 00:04:43.579
And this is a little bit
more of an interesting thing,

00:04:43.579 --> 00:04:46.440
because we know, first
of all, that there

00:04:46.440 --> 00:04:49.912
are 52 possibilities.

00:04:49.912 --> 00:04:51.370
But how many of
those possibilities

00:04:51.370 --> 00:04:57.010
meet these conditions that
it is a Jack or a heart.

00:04:57.010 --> 00:05:00.080
And to understand that,
I'll draw a Venn diagram.

00:05:00.080 --> 00:05:02.650
Sounds kind of fancy,
but nothing fancy here.

00:05:02.650 --> 00:05:04.740
So imagine that this
rectangle I'm drawing here

00:05:04.740 --> 00:05:06.820
represents all of the outcomes.

00:05:06.820 --> 00:05:09.770
So if you want, you could
imagine it has an area of 52.

00:05:09.770 --> 00:05:13.730
So this is 52 possible outcomes.

00:05:13.730 --> 00:05:16.730
Now, how many of those
outcomes result in a Jack?

00:05:16.730 --> 00:05:20.860
So we already learned, one out
of 13 of those outcomes result

00:05:20.860 --> 00:05:21.860
in a Jack.

00:05:21.860 --> 00:05:24.490
So I could draw a
little circle here,

00:05:24.490 --> 00:05:27.290
where that area-- and I'm
approximating-- represents

00:05:27.290 --> 00:05:28.520
the probability of a Jack.

00:05:28.520 --> 00:05:31.770
So it should be
roughly 1/13, or 4/52,

00:05:31.770 --> 00:05:33.460
of this area right over here.

00:05:33.460 --> 00:05:36.060
So I'll just draw it like this.

00:05:36.060 --> 00:05:38.505
So this right over here is
the probability of a Jack.

00:05:44.310 --> 00:05:46.660
There's four possible
cards out of the 52.

00:05:46.660 --> 00:05:51.560
So that is 4/52,
or one out of 13.

00:05:54.270 --> 00:05:56.470
Now, what's the probability
of getting a hearts?

00:05:56.470 --> 00:05:58.540
Well, I'll draw another
little circle here

00:05:58.540 --> 00:05:59.530
that represents that.

00:05:59.530 --> 00:06:03.540
13 out of 52 cards
represent a heart.

00:06:03.540 --> 00:06:05.880
And actually, one of those
represents both a heart

00:06:05.880 --> 00:06:06.920
and a Jack.

00:06:06.920 --> 00:06:09.060
So I'm actually going
to overlap them,

00:06:09.060 --> 00:06:12.870
and hopefully this will
make sense in a second.

00:06:12.870 --> 00:06:18.010
So there's actually 13
cards that are a heart.

00:06:18.010 --> 00:06:19.530
So this is the number of hearts.

00:06:22.180 --> 00:06:25.110
And actually, let me write this
top thing that way as well.

00:06:25.110 --> 00:06:27.380
It makes it a little bit
clearer that we're actually

00:06:27.380 --> 00:06:32.320
looking at the number of Jacks.

00:06:37.390 --> 00:06:39.210
And of course,
this overlap right

00:06:39.210 --> 00:06:42.920
here is the number of Jacks
and hearts-- the number

00:06:42.920 --> 00:06:45.470
of items out of this 52 that
are both a Jack and a heart--

00:06:45.470 --> 00:06:47.160
it is in both sets here.

00:06:47.160 --> 00:06:50.810
It is in this green circle and
it is in this orange circle.

00:06:50.810 --> 00:06:53.500
So this right over here--
let me do that in yellow

00:06:53.500 --> 00:06:56.600
since I did that problem in
yellow-- this right over here

00:06:56.600 --> 00:06:58.210
is a number of Jacks and hearts.

00:06:58.210 --> 00:06:59.710
So let me draw a
little arrow there.

00:06:59.710 --> 00:07:01.590
It's getting a little
cluttered, maybe

00:07:01.590 --> 00:07:03.325
I should draw a little
bit bigger number.

00:07:11.040 --> 00:07:12.960
And that's an
overlap over there.

00:07:12.960 --> 00:07:15.670
So what is the probability
of getting a Jack or a heart?

00:07:15.670 --> 00:07:18.450
So if you think about
it, the probability

00:07:18.450 --> 00:07:20.260
is going to be the
number of events

00:07:20.260 --> 00:07:22.977
that meet these conditions,
over the total number events.

00:07:22.977 --> 00:07:25.060
We already know the total
number of events are 52.

00:07:25.060 --> 00:07:27.040
But how many meet
these conditions?

00:07:27.040 --> 00:07:29.950
So it's going to be the
number-- you could say,

00:07:29.950 --> 00:07:32.110
well, look at the green
circle right there says

00:07:32.110 --> 00:07:35.580
the number that gives us a Jack,
and the orange circle tells us

00:07:35.580 --> 00:07:37.580
the number that
gives us a heart.

00:07:37.580 --> 00:07:41.100
So you might want to say,
well, why don't we add up

00:07:41.100 --> 00:07:43.800
the green and the
orange, but if you

00:07:43.800 --> 00:07:46.047
did that, you would
be double counting,

00:07:46.047 --> 00:07:47.630
Because if you add
it up-- if you just

00:07:47.630 --> 00:07:52.580
did four plus 13--
what are we saying?

00:07:52.580 --> 00:07:55.980
We're saying that
there are four Jacks

00:07:55.980 --> 00:08:00.170
and we're saying that
there are 13 hearts.

00:08:00.170 --> 00:08:04.230
But in both of these, when we
do it this way, in both cases

00:08:04.230 --> 00:08:06.239
we are counting
the Jack of hearts.

00:08:06.239 --> 00:08:07.780
We're putting the
Jack of hearts here

00:08:07.780 --> 00:08:09.370
and we're putting the
Jack of hearts here.

00:08:09.370 --> 00:08:10.869
So we're counting
the Jack of hearts

00:08:10.869 --> 00:08:13.890
twice, even though there's
only one card there.

00:08:13.890 --> 00:08:17.090
So you would have to subtract
out where they're common.

00:08:17.090 --> 00:08:21.090
You would have to
subtract out the item that

00:08:21.090 --> 00:08:23.390
is both a Jack and a heart.

00:08:23.390 --> 00:08:24.990
So you would subtract out a 1.

00:08:24.990 --> 00:08:26.800
Another way to
think about it is,

00:08:26.800 --> 00:08:29.130
you really want to figure
out the total area here.

00:08:33.919 --> 00:08:36.590
And let me zoom in-- and I'll
generalize it a little bit.

00:08:36.590 --> 00:08:38.090
So if you have one
circle like that,

00:08:38.090 --> 00:08:41.030
and then you have another
overlapping circle like that,

00:08:41.030 --> 00:08:43.730
and you wanted to figure
out the total area of both

00:08:43.730 --> 00:08:45.700
of these circles
combined, you would

00:08:45.700 --> 00:08:47.260
look at the area of this circle.

00:08:50.340 --> 00:08:53.730
And then you could add it
to the area of this circle.

00:08:53.730 --> 00:08:56.530
But when you do that, you'll
see that when you add the two

00:08:56.530 --> 00:08:59.270
areas, you're counting
this area twice.

00:08:59.270 --> 00:09:01.270
So in order to only
count that area once,

00:09:01.270 --> 00:09:04.510
you have to subtract
that area from the sum.

00:09:04.510 --> 00:09:10.200
So if this area has
A, this area is B,

00:09:10.200 --> 00:09:15.830
and the intersection
where they overlap is C,

00:09:15.830 --> 00:09:20.370
the combined area is
going to be A plus B-- --

00:09:20.370 --> 00:09:23.492
minus where they
overlap-- minus C.

00:09:23.492 --> 00:09:24.950
So that's the same
thing over here,

00:09:24.950 --> 00:09:26.874
we're counting all
the Jacks, and that

00:09:26.874 --> 00:09:28.040
includes the Jack of hearts.

00:09:28.040 --> 00:09:29.664
We're counting all
the hearts, and that

00:09:29.664 --> 00:09:31.230
includes the Jack of hearts.

00:09:31.230 --> 00:09:33.250
So we counted the
Jack of hearts twice,

00:09:33.250 --> 00:09:35.260
so we have to subtract
1 out of that.

00:09:35.260 --> 00:09:37.640
This is going to be
4 plus 13 minus 1,

00:09:37.640 --> 00:09:40.284
or this is going to be 16/52.

00:09:42.850 --> 00:09:48.250
And both of these things
are divisible by 4.

00:09:48.250 --> 00:09:51.890
So this is going to be the
same thing as, divide 16 by 4,

00:09:51.890 --> 00:09:52.670
you get 4.

00:09:52.670 --> 00:09:55.330
52 divided by 4 is 13.

00:09:55.330 --> 00:10:01.460
So there's a 4/13 chance that
you'd get a Jack or a hearts.


Probability explained

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:02.460
What I want to do in
this video is give you

00:00:02.460 --> 00:00:06.420
at least a basic
overview of probability.

00:00:06.420 --> 00:00:09.410
Probability, a word that
you've probably heard a lot of,

00:00:09.410 --> 00:00:11.640
and you are probably a
little bit familiar with it.

00:00:11.640 --> 00:00:12.520
But hopefully,
this will give you

00:00:12.520 --> 00:00:13.769
a little deeper understanding.

00:00:16.530 --> 00:00:19.719
Let's say that I have
a fair coin over here.

00:00:19.719 --> 00:00:21.260
And so when I talk
about a fair coin,

00:00:21.260 --> 00:00:23.290
I mean that it has
an equal chance

00:00:23.290 --> 00:00:26.190
of landing on one
side or another.

00:00:26.190 --> 00:00:28.820
So you can maybe view it
as the sides are equal,

00:00:28.820 --> 00:00:31.370
their weight is the
same on either side.

00:00:31.370 --> 00:00:33.270
If I flip it in
the air, it's not

00:00:33.270 --> 00:00:35.390
more likely to land on
one side or the other.

00:00:35.390 --> 00:00:37.110
It's equally likely.

00:00:37.110 --> 00:00:38.970
And so you have one
side of this coin.

00:00:38.970 --> 00:00:42.170
So this would be
the heads I guess.

00:00:42.170 --> 00:00:43.420
Try to draw George Washington.

00:00:43.420 --> 00:00:45.700
I'll assume it's a
quarter of some kind.

00:00:45.700 --> 00:00:48.210
And the other side, of
course, is the tails.

00:00:48.210 --> 00:00:49.810
So that is heads.

00:00:49.810 --> 00:00:52.490
The other side right
over there is tails.

00:00:52.490 --> 00:00:54.280
And so if I were
to ask you, what

00:00:54.280 --> 00:00:58.710
is the probability-- I'm
going to flip a coin.

00:00:58.710 --> 00:01:02.350
And I want to know what is the
probability of getting heads.

00:01:02.350 --> 00:01:03.780
And I could write
that like this--

00:01:03.780 --> 00:01:07.270
the probability
of getting heads.

00:01:07.270 --> 00:01:09.250
And you probably, just
based on that question,

00:01:09.250 --> 00:01:11.450
have a sense of what
probability is asking.

00:01:11.450 --> 00:01:17.300
It's asking for some
type of way of getting

00:01:17.300 --> 00:01:20.320
your hands around an event
that's fundamentally random.

00:01:20.320 --> 00:01:22.640
We don't know whether
it's heads or tails,

00:01:22.640 --> 00:01:25.890
but we can start to
describe the chances of it

00:01:25.890 --> 00:01:27.060
being heads or tails.

00:01:27.060 --> 00:01:30.117
And we'll talk about different
ways of describing that.

00:01:30.117 --> 00:01:31.700
So one way to think
about it, and this

00:01:31.700 --> 00:01:33.240
is the way that
probability tends

00:01:33.240 --> 00:01:36.130
to be introduced in textbooks,
is you say, well, look,

00:01:36.130 --> 00:01:40.110
how many different, equally
likely possibilities are there?

00:01:40.110 --> 00:01:42.800
So how many equally
likely possibilities.

00:01:42.800 --> 00:01:49.040
So number of equally--
let me write equally--

00:01:49.040 --> 00:01:53.006
of equally likely possibilities.

00:01:56.670 --> 00:01:59.530
And of the number of
equally possibilities,

00:01:59.530 --> 00:02:04.630
I care about the number that
contain my event right here.

00:02:04.630 --> 00:02:13.200
So the number of possibilities
that meet my constraint,

00:02:13.200 --> 00:02:16.680
that meet my conditions.

00:02:19.650 --> 00:02:21.920
So in the case of the
probability of figuring out

00:02:21.920 --> 00:02:25.970
heads, what is the number of
equally likely possibilities?

00:02:25.970 --> 00:02:28.100
Well, there's only
two possibilities.

00:02:28.100 --> 00:02:30.420
We're assuming that the coin
can't land on its corner

00:02:30.420 --> 00:02:31.545
and just stand straight up.

00:02:31.545 --> 00:02:33.700
We're assuming
that it lands flat.

00:02:33.700 --> 00:02:35.490
So there's two
possibilities here,

00:02:35.490 --> 00:02:37.670
two equally likely
possibilities.

00:02:37.670 --> 00:02:40.457
You could either get heads,
or you could get tails.

00:02:40.457 --> 00:02:42.040
And what's the number
of possibilities

00:02:42.040 --> 00:02:43.540
that meet my conditions?

00:02:43.540 --> 00:02:46.600
Well, there's only one,
the condition of heads.

00:02:46.600 --> 00:02:49.280
So it'll be 1/2.

00:02:49.280 --> 00:02:52.730
So one way to think about it
is the probability of getting

00:02:52.730 --> 00:02:56.730
heads is equal to 1/2.

00:02:56.730 --> 00:02:59.160
If I wanted to write
that as a percentage,

00:02:59.160 --> 00:03:03.590
we know that 1/2 is
the same thing as 50%.

00:03:03.590 --> 00:03:07.660
Now, another way to think about
or conceptualize probability

00:03:07.660 --> 00:03:10.070
that will give you
this exact same answer

00:03:10.070 --> 00:03:14.160
is to say, well, if I were to
run the experiment of flipping

00:03:14.160 --> 00:03:17.369
a coin-- so this flip, you
view this as an experiment.

00:03:17.369 --> 00:03:19.910
I know this isn't the kind of
experiment that you're used to.

00:03:19.910 --> 00:03:22.940
You know, you normally think an
experiment is doing something

00:03:22.940 --> 00:03:25.200
in chemistry or physics
or all the rest.

00:03:25.200 --> 00:03:26.790
But an experiment
is every time you

00:03:26.790 --> 00:03:29.390
do, you run this random event.

00:03:29.390 --> 00:03:30.950
So one way to think
about probability

00:03:30.950 --> 00:03:33.910
is if I were to do this
experiment, an experiment

00:03:33.910 --> 00:03:36.960
many, many, many times-- if
I were to do it 1,000 times

00:03:36.960 --> 00:03:39.880
or a million times or a billion
times or a trillion times--

00:03:39.880 --> 00:03:42.280
and the more the better--
what percentage of

00:03:42.280 --> 00:03:44.600
those would give me
what I care about?

00:03:44.600 --> 00:03:47.260
What percentage of those
would give me heads?

00:03:47.260 --> 00:03:50.520
And so another way to think
about this 50% probability

00:03:50.520 --> 00:03:54.020
of getting heads is if I
were to run this experiment

00:03:54.020 --> 00:03:58.370
tons of times, if I were
to run this forever,

00:03:58.370 --> 00:04:00.850
an infinite number of times,
what percentage of those

00:04:00.850 --> 00:04:01.810
would be heads?

00:04:01.810 --> 00:04:03.075
You would get this 50%.

00:04:03.075 --> 00:04:04.750
And you can run that simulation.

00:04:04.750 --> 00:04:05.860
You can flip a coin.

00:04:05.860 --> 00:04:07.360
And it's actually
a fun thing to do.

00:04:07.360 --> 00:04:08.520
I encourage you to do it.

00:04:08.520 --> 00:04:12.230
If you take 100 or 200
quarters or pennies,

00:04:12.230 --> 00:04:14.640
stick them in a big
box, shake the box

00:04:14.640 --> 00:04:18.170
so you're kind of simultaneously
flipping all of the coins,

00:04:18.170 --> 00:04:20.579
and then count how many of
those are going to be heads.

00:04:20.579 --> 00:04:24.302
And you're going to see that the
larger the number that you are

00:04:24.302 --> 00:04:25.760
doing, the more
likely you're going

00:04:25.760 --> 00:04:27.690
to get something
really close to 50%.

00:04:27.690 --> 00:04:30.210
And there's always some
chance-- even if you flipped

00:04:30.210 --> 00:04:33.960
a coin a million times, there's
some super-duper small chance

00:04:33.960 --> 00:04:35.210
that you would get all tails.

00:04:35.210 --> 00:04:38.740
But the more you
do, the more likely

00:04:38.740 --> 00:04:41.860
that things are going to
trend towards 50% of them

00:04:41.860 --> 00:04:43.510
are going to be heads.

00:04:43.510 --> 00:04:45.850
Now, let's just apply
these same ideas.

00:04:45.850 --> 00:04:48.790
And while we're starting with
probability, at least kind

00:04:48.790 --> 00:04:51.130
of the basic, this is
probably an easier thing

00:04:51.130 --> 00:04:52.030
to conceptualize.

00:04:52.030 --> 00:04:54.363
But a lot of times, this is
actually a helpful one, too,

00:04:54.363 --> 00:04:56.800
this idea that if you run the
experiment many, many, many,

00:04:56.800 --> 00:04:59.656
many times, what
percentage of those trials

00:04:59.656 --> 00:05:01.530
are going to give you
what you're asking for.

00:05:01.530 --> 00:05:03.200
In this case, it was heads.

00:05:03.200 --> 00:05:05.546
Now, let's do another
very typical example

00:05:05.546 --> 00:05:06.920
when you first
learn probability.

00:05:06.920 --> 00:05:10.210
And this is the idea
of rolling a die.

00:05:10.210 --> 00:05:13.240
So here's my die
right over here.

00:05:13.240 --> 00:05:16.650
And of course, you have, you
know, the different sides

00:05:16.650 --> 00:05:17.170
of the die.

00:05:17.170 --> 00:05:18.570
So that's the 1.

00:05:18.570 --> 00:05:19.560
That's the 2.

00:05:19.560 --> 00:05:21.682
And that's the 3.

00:05:21.682 --> 00:05:23.640
And what I want to do--
and we know, of course,

00:05:23.640 --> 00:05:27.010
that there are-- and I'm
assuming this is a fair die.

00:05:27.010 --> 00:05:30.070
And so there are six equally
likely possibilities.

00:05:30.070 --> 00:05:34.600
When you roll this, you could
get a 1, a 2, a 3, a 4, a 5,

00:05:34.600 --> 00:05:35.120
or a 6.

00:05:35.120 --> 00:05:37.550
And they're all equally likely.

00:05:37.550 --> 00:05:42.340
So if I were to ask you,
what is the probability

00:05:42.340 --> 00:05:45.390
given that I'm rolling a fair
die-- so the experiment is

00:05:45.390 --> 00:05:50.860
rolling this fair die, what is
the probability of getting a 1?

00:05:50.860 --> 00:05:54.430
Well, what are the number of
equally likely possibilities?

00:05:54.430 --> 00:05:57.440
Well, I have six equally
likely possibilities.

00:05:57.440 --> 00:05:59.900
And how many of those
meet my conditions?

00:05:59.900 --> 00:06:03.150
Well, only one of them meets
my condition, that right there.

00:06:03.150 --> 00:06:07.650
So there is a 1/6
probability of rolling a 1.

00:06:07.650 --> 00:06:16.890
What is the probability
of rolling a 1 or a 6?

00:06:16.890 --> 00:06:22.190
Well, once again, there are six
equally likely possibilities

00:06:22.190 --> 00:06:24.187
for what I can get.

00:06:24.187 --> 00:06:26.520
There are now two possibilities
that meet my conditions.

00:06:26.520 --> 00:06:31.620
I could roll a 1 or
I could roll a 6.

00:06:31.620 --> 00:06:35.220
So now there are
two possibilities

00:06:35.220 --> 00:06:38.150
that meet my constraints,
my conditions.

00:06:38.150 --> 00:06:42.630
There is a 1/3 probability
of rolling a 1 or a 6.

00:06:42.630 --> 00:06:45.534
Now, what is the
probability-- and this

00:06:45.534 --> 00:06:47.700
might seem a little silly
to even ask this question,

00:06:47.700 --> 00:06:49.590
but I'll ask it just
to make it clear.

00:06:49.590 --> 00:06:55.430
What is the probability
of rolling a 2 and a 3?

00:06:55.430 --> 00:06:57.930
And I'm just talking
about one roll of the die.

00:06:57.930 --> 00:07:01.920
Well, in any roll of the die,
I can only get a 2 or a 3.

00:07:01.920 --> 00:07:04.890
I'm not talking about taking
two rolls of this die.

00:07:04.890 --> 00:07:09.040
So in this situation,
there's six possibilities,

00:07:09.040 --> 00:07:12.570
but none of these
possibilities are 2 and a 3.

00:07:12.570 --> 00:07:14.440
None of these are 2 and a 3.

00:07:14.440 --> 00:07:16.250
2 and a 3 cannot exist.

00:07:16.250 --> 00:07:23.390
On one trial, you cannot get a 2
and a 3 in the same experiment.

00:07:23.390 --> 00:07:26.470
Getting a 2 and a 3 are
mutually exclusive events.

00:07:26.470 --> 00:07:29.130
They cannot happen
at the same time.

00:07:29.130 --> 00:07:31.432
So the probability of
this is actually 0.

00:07:31.432 --> 00:07:33.890
There's no way to roll this
normal die and all of a sudden,

00:07:33.890 --> 00:07:35.098
you get a 2 and a 3, in fact.

00:07:35.098 --> 00:07:38.060
And I don't want to confuse
you with that, because it's

00:07:38.060 --> 00:07:39.820
kind of abstract and impossible.

00:07:39.820 --> 00:07:43.130
So let's cross this
out right over here.

00:07:43.130 --> 00:07:50.160
Now, what is the probability
of getting an even number?

00:07:50.160 --> 00:07:53.450
So once again, you have six
equally likely possibilities

00:07:53.450 --> 00:07:54.980
when I roll that die.

00:07:54.980 --> 00:07:58.980
And which of these possibilities
meet my conditions,

00:07:58.980 --> 00:08:01.450
the condition of being even?

00:08:01.450 --> 00:08:06.110
Well, 2 is even, 4 is
even, and 6 is even.

00:08:06.110 --> 00:08:09.780
So 3 of the possibilities
meet my conditions,

00:08:09.780 --> 00:08:11.350
meet my constraints.

00:08:11.350 --> 00:08:12.600
So this is 1/2.

00:08:12.600 --> 00:08:15.310
If I roll a die, I
have a 1/2 chance

00:08:15.310 --> 00:08:18.130
of getting an even number.


New operator definitions

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:03.090
We're all used to the
traditional operators

00:00:03.090 --> 00:00:06.810
like addition and subtraction
and multiplication

00:00:06.810 --> 00:00:07.600
and division.

00:00:07.600 --> 00:00:10.350
And we've seen there's multiple
ways to represent this.

00:00:10.350 --> 00:00:14.490
But what we're going to do in
this video is a little fun.

00:00:14.490 --> 00:00:16.690
We're actually going to
define our own operators.

00:00:16.690 --> 00:00:18.370
And what's neat
about this is it kind

00:00:18.370 --> 00:00:20.051
of shows how broad
mathematics can be.

00:00:20.051 --> 00:00:22.300
And on a more practical
sense, it's actually something

00:00:22.300 --> 00:00:24.234
that you might see on
some standardized tests.

00:00:24.234 --> 00:00:25.650
And the reason why
they do that is

00:00:25.650 --> 00:00:27.040
so that you can appreciate
that these aren't

00:00:27.040 --> 00:00:29.730
the only operators out there--
plus exponentiation and all

00:00:29.730 --> 00:00:31.380
those-- that in
mathematics, you can

00:00:31.380 --> 00:00:33.370
define a whole new
set of operators.

00:00:33.370 --> 00:00:34.680
So let's just do that.

00:00:34.680 --> 00:00:40.160
So let me just
define x diamond y.

00:00:40.160 --> 00:00:46.720
And I'm going to define
that as 5x minus y.

00:00:46.720 --> 00:00:48.970
So you could view this as
defining it a function.

00:00:48.970 --> 00:00:51.660
But we're defining
it using an operator.

00:00:51.660 --> 00:00:54.540
So if I have x diamond
y, by definition,

00:00:54.540 --> 00:00:55.880
we have defined this operator.

00:00:55.880 --> 00:00:59.480
That means that's going
to be equal to 5x minus y.

00:00:59.480 --> 00:01:07.720
So given that definition,
what would 7 diamond 11 be?

00:01:07.720 --> 00:01:09.240
Well, you just go
to the definition.

00:01:09.240 --> 00:01:11.950
7 diamond 11.

00:01:11.950 --> 00:01:13.600
Instead of an x we have a 7.

00:01:13.600 --> 00:01:15.990
So it's going to be 5 times 7.

00:01:15.990 --> 00:01:22.040
So let me do it 5 times 7
minus, and instead of a y,

00:01:22.040 --> 00:01:23.510
we have an 11.

00:01:23.510 --> 00:01:26.450
So one way to think about
it is, in our definition,

00:01:26.450 --> 00:01:28.690
every place you saw an x,
you could replace with a 7,

00:01:28.690 --> 00:01:31.210
every place you saw a y,
you replace with an 11.

00:01:31.210 --> 00:01:34.030
So you have minus 11 over here.

00:01:34.030 --> 00:01:36.700
Let me make the number--
So this is the 7.

00:01:36.700 --> 00:01:39.100
This 7 is this 7.

00:01:39.100 --> 00:01:44.840
And this 11 is this
11 right over here.

00:01:44.840 --> 00:01:46.230
And then we just evaluate that.

00:01:46.230 --> 00:01:48.500
So 5 times 7 is 35.

00:01:48.500 --> 00:01:53.870
So this is equal to 35 minus
11, which is equal to 24.

00:01:53.870 --> 00:02:00.190
So 7 diamond 11 is equal to 24.

00:02:00.190 --> 00:02:01.610
We can define other things.

00:02:01.610 --> 00:02:06.920
We can define something crazy
like, let me define a-- well,

00:02:06.920 --> 00:02:11.535
I mentioned a star, let me
use a star-- a star-- let

00:02:11.535 --> 00:02:15.240
me write it this way-- a star b.

00:02:15.240 --> 00:02:17.200
Let's say that that
is the same thing

00:02:17.200 --> 00:02:23.120
as-- I don't know--
a over a plus b.

00:02:23.120 --> 00:02:24.490
And so same idea.

00:02:24.490 --> 00:02:33.000
What would 5 star 6 be?

00:02:33.000 --> 00:02:34.662
Well, you go back
to the definition.

00:02:34.662 --> 00:02:36.620
By definition, every
place where you see the a,

00:02:36.620 --> 00:02:38.110
you would now replace with a 5.

00:02:38.110 --> 00:02:41.090
Every time you saw the b, you
would now replace with a 6.

00:02:41.090 --> 00:02:48.720
So this is going to be equal
to 5 over 5 plus 6. a plus b.

00:02:48.720 --> 00:02:52.440
a is 5, b is 6 over 5 plus 6.

00:02:52.440 --> 00:02:53.436
So this would be 5/11.

00:02:56.080 --> 00:02:57.430
And then you can compound them.

00:02:57.430 --> 00:02:59.430
And we haven't defined
any order of operations

00:02:59.430 --> 00:03:02.670
for these particular operations
that we've just defined.

00:03:02.670 --> 00:03:04.760
So we're going to be
careful to use parentheses

00:03:04.760 --> 00:03:07.420
when we put some
of these together,

00:03:07.420 --> 00:03:10.000
but you can do something
like, something interesting,

00:03:10.000 --> 00:03:22.430
like negative 1
diamond 0 star 5.

00:03:22.430 --> 00:03:24.530
And once again, we just
focus on parentheses,

00:03:24.530 --> 00:03:26.030
because that's the
only thing that's

00:03:26.030 --> 00:03:27.446
telling us what
to start on first.

00:03:27.446 --> 00:03:29.060
Because we haven't
figured out, we

00:03:29.060 --> 00:03:32.246
haven't defined whether diamond
takes precedence over star,

00:03:32.246 --> 00:03:34.120
or star takes precedence
over diamond the way

00:03:34.120 --> 00:03:36.530
that we have that saying that,
hey, you do multiplication

00:03:36.530 --> 00:03:37.370
before you do addition.

00:03:37.370 --> 00:03:39.161
We haven't defined it
for those operations,

00:03:39.161 --> 00:03:41.250
but that's what the
parentheses helps us do.

00:03:41.250 --> 00:03:43.540
So we want to evaluate
these parentheses first.

00:03:43.540 --> 00:03:49.170
0 star 5 that is 0-- because
you could view this 0

00:03:49.170 --> 00:03:55.120
as the a and the 5 as the b--so
it's going to be 0 over 0 plus

00:03:55.120 --> 00:04:00.480
5, which is just going to be 0.

00:04:00.480 --> 00:04:04.160
So this over here, 0
or 5 just goes to 0.

00:04:04.160 --> 00:04:06.430
So this whole
expression simplifies

00:04:06.430 --> 00:04:09.490
to negative 1 diamond--
this diamond right

00:04:09.490 --> 00:04:12.480
over here-- diamond 0.

00:04:12.480 --> 00:04:15.360
And now we go to the definition
of the diamond operator.

00:04:15.360 --> 00:04:19.390
Well, that's five times the
first number in our operator,

00:04:19.390 --> 00:04:21.700
or the first term that
we're giving the operator.

00:04:21.700 --> 00:04:23.580
I guess you could
think of it that way.

00:04:23.580 --> 00:04:25.090
So 5 times that.

00:04:25.090 --> 00:04:30.690
So it'll be 5 times negative
1, x is negative 1 minus y.

00:04:30.690 --> 00:04:32.700
Well, y here is the 0.

00:04:32.700 --> 00:04:33.720
Minus 0.

00:04:33.720 --> 00:04:39.430
So 5 times negative
1 is negative 5.

00:04:39.430 --> 00:04:41.870
And you will see--
and the idea here

00:04:41.870 --> 00:04:44.460
is just to make you feel
comfortable defining

00:04:44.460 --> 00:04:45.460
new operators like this.

00:04:45.460 --> 00:04:47.919
And not being daunted if all
of a sudden you see a diamond,

00:04:47.919 --> 00:04:49.626
and they're defining
the diamond for you.

00:04:49.626 --> 00:04:51.560
And you're like, wait,
I never saw a diamond.

00:04:51.560 --> 00:04:53.300
They're actually defining it
for you, so you shouldn't say,

00:04:53.300 --> 00:04:54.190
I never saw a diamond.

00:04:54.190 --> 00:04:56.481
You should say, well, they've
defined a diamond for me.

00:04:56.481 --> 00:04:58.270
This is how I use that operator.

00:04:58.270 --> 00:05:00.670
And sometimes you'll
see even wackier things.

00:05:00.670 --> 00:05:05.130
You'll see things like this.

00:05:05.130 --> 00:05:06.420
Let me draw.

00:05:06.420 --> 00:05:09.050
So they'll define.

00:05:09.050 --> 00:05:11.429
I don't know if you would even
consider this an operator.

00:05:11.429 --> 00:05:13.720
But you'll see something like
this, that by definition,

00:05:13.720 --> 00:05:15.700
if someone writes
a symbol like this,

00:05:15.700 --> 00:05:23.430
and they put-- a, b, c-- let me
write it this way-- a, b, c, d.

00:05:23.430 --> 00:05:26.074
They'll say this is
the same thing as ad

00:05:26.074 --> 00:05:29.040
minus b, all of that over c.

00:05:29.040 --> 00:05:31.230
And once again, this
is just a definition.

00:05:31.230 --> 00:05:33.120
They have this
weird symbolic way

00:05:33.120 --> 00:05:35.420
of representing these
variables in all this.

00:05:35.420 --> 00:05:39.820
But they're defining how do you
evaluate this crazy expression.

00:05:39.820 --> 00:05:42.750
And so, if someone
were to give you,

00:05:42.750 --> 00:05:46.830
were to say, evaluate
this diamond.

00:05:46.830 --> 00:05:48.720
Let me evaluate the diamond.

00:05:48.720 --> 00:05:53.030
So evaluate the diamond
where, in my little sections

00:05:53.030 --> 00:05:58.350
of the diamond, I have a
negative 1, a 5, a 3, and a 2.

00:05:58.350 --> 00:06:00.070
We would just use
the definition of how

00:06:00.070 --> 00:06:01.702
to evaluate this diamond.

00:06:01.702 --> 00:06:03.160
And we'd say, OK,
every time we see

00:06:03.160 --> 00:06:05.010
an a, that is going
to be negative 1.

00:06:05.010 --> 00:06:09.330
So we have a negative 1 times d.

00:06:09.330 --> 00:06:11.880
Well, d is whatever
is in the bottom right

00:06:11.880 --> 00:06:14.080
section of this
diameter or this kite.

00:06:14.080 --> 00:06:15.109
So d is going to be 2.

00:06:15.109 --> 00:06:16.150
Let me write it this way.

00:06:16.150 --> 00:06:16.970
This is a.

00:06:16.970 --> 00:06:18.330
This is b.

00:06:18.330 --> 00:06:19.820
This is c.

00:06:19.820 --> 00:06:21.700
And this is d.

00:06:21.700 --> 00:06:26.970
So it's going to be negative
1 times 2 minus b-- well b

00:06:26.970 --> 00:06:32.240
is 5-- minus 5, all of
that over c, which is 3.

00:06:32.240 --> 00:06:35.320
So this is going to be
equal to negative 2 minus 5.

00:06:35.320 --> 00:06:40.090
So that is negative 7 over 3.

00:06:40.090 --> 00:06:41.580
And you could go
crazy like this.

00:06:41.580 --> 00:06:43.163
And it might be a
fun thing, actually,

00:06:43.163 --> 00:06:44.400
if you have some spare time.

00:06:44.400 --> 00:06:46.750
Define your own operators
and see how creative

00:06:46.750 --> 00:06:49.870
you can get with
those operators.


Heart disease and heart attacks

WEBVTT
Kind: captions
Language: en

00:00:00.830 --> 00:00:04.350
I think most of us know what the heart does in our body.

00:00:04.360 --> 00:00:09.790
It pumps the blood and in particular it takes in the blood from the rest of the body.

00:00:09.790 --> 00:00:14.800
That is blue over here in this diagram, because that blood does not have carried oxygen

00:00:14.810 --> 00:00:20.030
And it pumps that blood to the lungs where the blood gets oxygenated.

00:00:20.190 --> 00:00:27.420
And then it comes back from the lungs, depicted as red blood, now that it has oxygen in it.

00:00:27.420 --> 00:00:32.040
And then that oxygenated blood is pumped to the rest of the body.

00:00:32.040 --> 00:00:38.140
And I think also, most of us have the general idea that when people talk about heart disease or heart attacks,

00:00:38.140 --> 00:00:45.490
which we learn, are two different things, related but different things, that has something to do with the clogging of arteries.

00:00:45.500 --> 00:00:50.280
so when I was a kid and I first learnt about the clogging of arteries.

00:00:50.290 --> 00:00:53.550
and I knew enough that the heart is all about pumping blood to the body.

00:00:53.560 --> 00:01:00.340
I assumed the artery that people were talking about these big arteries that were coming away from the heart

00:01:00.340 --> 00:01:03.560
to the rest of the body that somehow these things got clogged up.

00:01:03.570 --> 00:01:07.160
so let me draw that, different colour you can see

00:01:07.170 --> 00:01:09.740
so this was my ....this is what I thought

00:01:09.750 --> 00:01:13.620
people were talking about when they were saying clogging of the arteries

00:01:13.620 --> 00:01:18.050
and maybe when they got clogged enough, the stopped blood flow to the rest of the body somehow

00:01:18.060 --> 00:01:23.870
and that would actually kill the person. I want to make it very clear right now.

00:01:23.870 --> 00:01:28.350
Those are not the arteries that people are talking about getting clogged,

00:01:28.360 --> 00:01:32.360
when people talk about heart disease or heart attacks.

00:01:32.360 --> 00:01:38.370
The arteries that they are talking about are the arteries that actually provide blood to the heart.

00:01:38.380 --> 00:01:42.930
Remember the heart itself is a muscle. It itself needs oxygen.

00:01:42.930 --> 00:01:51.170
So you have these arteries right over here, the red tubes. Those are arteries.

00:01:51.170 --> 00:01:52.880
and then the blue ones are veins.

00:01:52.890 --> 00:01:56.300
They're taking the de-oxygenated blood away from the tissue of the heart.

00:01:56.300 --> 00:01:58.390
And these are called coronary arteries.

00:01:58.390 --> 00:02:03.700
And this one over here at least from the point of view of me or you looks like it's on the right.

00:02:03.710 --> 00:02:08.670
Or from the point of view of the person whose heart it is, it's on the left.

00:02:08.680 --> 00:02:13.320
This right over here is called the left coronary artery or LCA.

00:02:13.320 --> 00:02:18.310
And this right over here in red is called the right coronary arteries or the RCA.

00:02:18.320 --> 00:02:23.310
And so when people talk about arteries getting blocked or getting clogged,

00:02:23.310 --> 00:02:25.930
they're talking about the coronary arteries.

00:02:25.930 --> 00:02:30.380
They're talking about the things that supply blood to the heart.

00:02:30.390 --> 00:02:40.320
So let's zoom in on one of them....Maybe we can zoom in right over here, that part of the artery.

00:02:40.330 --> 00:02:49.240
That's the tube....clear where I am zooming in. I am zooming in right over here.

00:02:49.250 --> 00:02:55.550
So over time, I am not going into the details how this happened. It is subject for another video.

00:02:55.550 --> 00:03:00.150
You can have these plaques build up along the walls of the artery.

00:03:00.160 --> 00:03:05.770
So over time if a person doesn't have the right diet, or maybe they just have a predisposition to it,

00:03:05.780 --> 00:03:11.350
you can have these things called plaques form on the walls of the arteries.

00:03:11.350 --> 00:03:15.760
And the plaques, the material inside of them are lipids, so things like

00:03:15.760 --> 00:03:19.150
fat, cholesterol and also dead white blood cells,

00:03:19.150 --> 00:03:28.730
which is this kind of messy substance right over here. This is what we call a plaque.

00:03:28.740 --> 00:03:35.190
And the formation of these plaques that obstruct the actual blood vessel, that actually obstruct the artery.

00:03:35.200 --> 00:03:44.070
We call it.....make it clear you see that. This is kind of tube over here. Let me draw the blood

00:03:44.070 --> 00:03:59.180
So this formation of these plaques we call atherosclerosis.

00:03:59.180 --> 00:04:03.860
So you can imagine if you have these things build up,

00:04:03.870 --> 00:04:10.810
it's narrowing the actual vessel that's supplying blood downstream from that obstruction.

00:04:10.820 --> 00:04:16.570
So it would be destructing the blood flow downstream right over there.

00:04:16.570 --> 00:04:23.000
In that general process we talk about the restriction of blood flow, that is ischemia that's happening.

00:04:23.010 --> 00:04:28.430
So ischemia is deprivation of blood flow and oxygen downstream from this right over there.

00:04:28.430 --> 00:04:39.400
That's what we call coronary artery disease, or heart disease.

00:04:39.410 --> 00:04:52.380
So this causes coronary artery disease, which is sometimes called heart disease.

00:04:52.380 --> 00:04:58.570
Coronary heart disease would be redundant, because coronary is already referring to the heart.

00:04:58.580 --> 00:05:07.170
This is also sometimes called heart disease. And so you can imagine if downstream the muscle tissue is

00:05:07.170 --> 00:05:09.970
not getting all the oxygen it needs, especially maybe when this person, whoever's heart this is,

00:05:09.980 --> 00:05:17.380
when they are exerting themselves, they need more oxygen. The heart needs to pump a little harder.

00:05:17.390 --> 00:05:20.960
If downstream the cells are not getting all of the oxygen they need,

00:05:20.970 --> 00:05:25.500
you can imagine that the heart maybe not able to provide all of the functions,

00:05:25.510 --> 00:05:29.340
whoever's heart this is, that they needed to do.

00:05:29.340 --> 00:05:33.770
And when that happens that's called heart failure.

00:05:33.780 --> 00:05:39.350
So heart disease is one of the causes of heart failure.

00:05:39.360 --> 00:05:43.460
Now I want to be clear, heart failure does not mean that the heart is stopping.

00:05:43.460 --> 00:05:48.710
That the heart is stopped and the person is dead, it literally just means

00:05:48.720 --> 00:05:53.550
that the heart is failing to do what it should be doing. It's failing to provide the needs of that person.

00:05:53.550 --> 00:06:01.280
So it's not pumping hard enough or well enough to provide adequate function for that person.

00:06:01.280 --> 00:06:06.670
The other symptom that actually might occur when someone has coronary heart disease,

00:06:06.680 --> 00:06:11.000
where they have obstruction or this ischemia

00:06:11.000 --> 00:06:18.700
or this deprivation of oxygen downstream from this obstruction, is that they might experience this kind of strangling chest pain

00:06:18.710 --> 00:06:23.260
That's called angina pectoris or sometimes angina.

00:06:23.260 --> 00:06:33.330
Very few people say the pectoris part. Sometimes you hear people say oh angina.

00:06:33.340 --> 00:06:40.260
Angina pectoris, which is really chest pain, angina literally comes from meaning this kind of strangling feeling,

00:06:40.260 --> 00:06:43.920
and pectoris is in the chest. This is strangling feeling in the chest.

00:06:43.930 --> 00:06:53.120
This is a symptom of heart disease. Now this is already not a situation you want to get into

00:06:53.120 --> 00:06:57.300
because already your body is not able to function as well as it maybe could be, as maybe it could.

00:06:57.300 --> 00:07:02.140
Sometimes as I might describe here is a plaque.

00:07:02.150 --> 00:07:06.910
Some plaques are actually unstable. These plaques just grow and grow and grow.

00:07:06.920 --> 00:07:12.250
It makes probably heart disease worse and worse, heart failure worse, and angina pectoris worse.

00:07:12.260 --> 00:07:15.740
These plaques are unstable. It can actually rupture.

00:07:15.740 --> 00:07:25.720
You can imagine all these blood flows, as plaque grows, the blood flows become a little turbulent around these plaques.

00:07:25.730 --> 00:07:29.130
It has to go really fast through this narrow section. It comes out turbulent on the inside and

00:07:29.130 --> 00:07:35.290
creates these friction and all the rest. At some point, you can imagine this plaque is unstable.

00:07:35.290 --> 00:07:45.590
It could actually rupture. Let me draw ruptured plaque over here....let me draw the same one....I am gonna draw the ruptured.

00:07:45.600 --> 00:07:54.700
Now this plaque has rupture. It got so big. Maybe the turbulent blood flow helped stimulate this and whatever else.

00:07:54.700 --> 00:07:59.520
For whatever reason, it ruptures.

00:07:59.530 --> 00:08:05.100
When it ruptures, now it sudden i am doing a simplification of the process right over here.

00:08:05.110 --> 00:08:13.510
Now the content of this plaque: the lipid, the cholesterol, the fat, the dead white blood cell.

00:08:13.510 --> 00:08:19.450
Now it sudden is exposed to the blood flow. in particular to the clotting factor in the blood.

00:08:19.460 --> 00:08:25.450
This is highly thrombogenic material. Thrombogenic, very fancy word.

00:08:25.450 --> 00:08:31.440
That just means, that tends to cause blood clot. Thrombosis is a blood clot.

00:08:31.440 --> 00:08:36.130
What happened is, as soon as this type of things happened, it can literally happen in seconds or in minutes.

00:08:36.130 --> 00:08:41.840
All the sudden you could have these clotting factors form a clot right over here right at the actual plaque.

00:08:41.840 --> 00:08:50.910
As this happens, it starts to really really really obstruct the blood vessel.

00:08:50.910 --> 00:08:54.730
Sometimes it can even completely obstruct the blood vessel.

00:08:54.740 --> 00:09:01.720
When it happens, you are significantly depleting the blood flow going downstream from there.

00:09:01.720 --> 00:09:12.790
You might even be shutting it off. When you do that, the cells downstream will no longer get oxygen.

00:09:12.790 --> 00:09:27.020
They will die. This right over here, is called an infarction. an infarct is actually a dead heart tissue.

00:09:27.030 --> 00:09:37.200
You can imagine when heart tissue begins to die, this is even worse than what we were describing with coronary heart disease.

00:09:37.200 --> 00:09:42.770
Not only coronary heart disease just is not getting enough oxygen, now they are actually dead.

00:09:42.770 --> 00:09:55.740
They are turning into dead tissue. This process of completely or almost completely depriving cells of oxygen so that they die, this is a heart attack.

00:09:55.750 --> 00:10:05.050
Let me completely obstruct this artery to make the point clear. this right over here is a heart attack.

00:10:05.060 --> 00:10:09.210
This is the primary cause of heart attack.

00:10:09.220 --> 00:10:16.200
It's less likely but sometimes a plaque could also go downstream, kind of form a thromboembolism.

00:10:16.210 --> 00:10:22.060
It would be this thrombogenic material, the clots around it.

00:10:22.060 --> 00:10:27.980
It would actually go and block the artery further downstream and be embolism.

00:10:27.980 --> 00:10:34.240
That can also block the artery and cause tissue to die. But the main cause is this intense clotting that can

00:10:34.250 --> 00:10:41.260
occur pretty quickly and completely obstruct the artery. There is one last word i want to touch

00:10:41.260 --> 00:10:52.220
sometimes mixed in with all the other words, that is cardiac arrest. That's because sometimes we use them in the same context.

00:10:52.220 --> 00:11:03.130
one thing can lead to another. Heart attack is not cardiac arrest. Cardiac arrest is the actual dying of the heart.

00:11:03.130 --> 00:11:12.915
What I just described in a heart attack, people can have a heart attack. They will have some part of their muscle tissue die.

00:11:12.915 --> 00:11:22.260
Some part infarct, that's what they called myocardial infarction. Myocardial means the tissue of heart or the muscle of the heart that's dying.

00:11:22.270 --> 00:11:27.370
Sometimes it is called myocardial infarction. That is not cardiac arrest.

00:11:27.370 --> 00:11:31.950
Cause you can have some of your heart tissue die and you can survive. Your heart would be impaired.

00:11:31.950 --> 00:11:37.660
But you will continue to live. Cardiac arrest is literally your heart stopping.

00:11:37.660 --> 00:11:41.830
This would obviously cause someone to die. If you have a bad enough heart attack,

00:11:41.840 --> 00:11:53.362
if you have enough of the tissue get starved of oxygen so that it dies, infarction occurs.

00:11:53.362 --> 00:11:57.980
Then it could lead to cardiac arrest. It always won't lead to cardiac arrest.

00:11:57.990 --> 00:12:02.810
Frankly, heart attack is not the only thing that can cause cardiac arrest.

00:12:02.810 --> 00:12:11.160
I also want to once again differentiate cardiac arrest from heart failure, because they sound the same, sound like heart is failing.

00:12:11.170 --> 99:59:59.999
Cardiac arrest is heart stopping. Heart failure is essentially just saying that heart can not provide all of the needs for the body.


Elements and atoms

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.120
We humans have known,
for thousands of years,

00:00:03.120 --> 00:00:05.341
just looking at our
environment around us,

00:00:05.341 --> 00:00:06.840
that there are
different substances.

00:00:06.840 --> 00:00:08.600
And these different
substances tend

00:00:08.600 --> 00:00:10.120
to have different properties.

00:00:10.120 --> 00:00:11.470
And not only do they have
different properties,

00:00:11.470 --> 00:00:13.400
one might reflect
light in a certain way,

00:00:13.400 --> 00:00:16.230
or not reflect light, or
be a certain color, or at

00:00:16.230 --> 00:00:20.390
a certain temperature, be
liquid or gas, or be a solid.

00:00:20.390 --> 00:00:22.290
But we also start
to observe how they

00:00:22.290 --> 00:00:25.030
react with each other in
certain circumstances.

00:00:25.030 --> 00:00:27.410
And here's pictures of
some of these substances.

00:00:27.410 --> 00:00:29.070
This right here is carbon.

00:00:29.070 --> 00:00:31.650
And this is in
its graphite form.

00:00:31.650 --> 00:00:33.910
This right here is lead.

00:00:33.910 --> 00:00:36.030
This right here is gold.

00:00:36.030 --> 00:00:38.750
And all of the ones that I've
shown pictures of, here--

00:00:38.750 --> 00:00:41.720
and I got them all from this
website, right over there--

00:00:41.720 --> 00:00:43.460
all of these are in
their solid form.

00:00:43.460 --> 00:00:45.830
But we also know that
it looks like there's

00:00:45.830 --> 00:00:49.240
certain types of air, and
certain types of air particles.

00:00:49.240 --> 00:00:51.090
And depending on what
type of air particles

00:00:51.090 --> 00:00:55.590
you're looking at, whether it
is carbon or oxygen or nitrogen,

00:00:55.590 --> 00:00:57.790
that seems to have different
types of properties.

00:00:57.790 --> 00:00:59.666
Or there are other things
that can be liquid.

00:00:59.666 --> 00:01:01.665
Or even if you raise the
temperature high enough

00:01:01.665 --> 00:01:02.360
on these things.

00:01:02.360 --> 00:01:04.485
You could, if you raise
the temperature high enough

00:01:04.485 --> 00:01:06.440
on gold or lead, you
could get a liquid.

00:01:06.440 --> 00:01:10.360
Or if you, kind of, if
you burn this carbon,

00:01:10.360 --> 00:01:12.020
you can get it to
a gaseous state.

00:01:12.020 --> 00:01:13.690
You can release it
into the atmosphere.

00:01:13.690 --> 00:01:14.880
You can break its structure.

00:01:14.880 --> 00:01:18.530
So these are things that we've
all, kind of, that humanity

00:01:18.530 --> 00:01:21.000
has observed for
thousands of years.

00:01:21.000 --> 00:01:22.530
But it leads to a
natural question

00:01:22.530 --> 00:01:24.420
that used to be a
philosophical question.

00:01:24.420 --> 00:01:26.790
But now we can answer
it a little bit better.

00:01:26.790 --> 00:01:31.320
And that question is, if you
keep breaking down this carbon,

00:01:31.320 --> 00:01:33.970
into smaller and
smaller chunks, is there

00:01:33.970 --> 00:01:38.930
some smallest chunk, some
smallest unit, of this stuff,

00:01:38.930 --> 00:01:43.314
of this substance, that still
has the properties of carbon?

00:01:43.314 --> 00:01:44.730
And if you were
to, somehow, break

00:01:44.730 --> 00:01:46.271
that even further,
somehow, you would

00:01:46.271 --> 00:01:48.270
lose the properties
of the carbon.

00:01:48.270 --> 00:01:50.380
And the answer is, there is.

00:01:50.380 --> 00:01:52.030
And so just to get
our terminology,

00:01:52.030 --> 00:01:54.940
we call these
different substances--

00:01:54.940 --> 00:01:58.070
these pure substances that
have these specific properties

00:01:58.070 --> 00:01:59.960
at certain
temperatures and react

00:01:59.960 --> 00:02:02.020
in certain ways-- we
call them elements.

00:02:05.210 --> 00:02:06.470
Carbon is an element.

00:02:06.470 --> 00:02:07.570
Lead is an element.

00:02:07.570 --> 00:02:08.910
Gold is an element.

00:02:08.910 --> 00:02:10.949
You might say that
water is an element.

00:02:10.949 --> 00:02:13.827
And in history, people
have referred to water

00:02:13.827 --> 00:02:14.410
as an element.

00:02:14.410 --> 00:02:18.250
But now we know that water is
made up of more basic elements.

00:02:18.250 --> 00:02:21.100
It's made of oxygen
and of hydrogen.

00:02:21.100 --> 00:02:25.120
And all of our elements
are listed here

00:02:25.120 --> 00:02:27.830
in the Periodic
Table of Elements.

00:02:27.830 --> 00:02:30.450
C stands for carbon-- I'm just
going through the ones that

00:02:30.450 --> 00:02:33.280
are very relevant to humanity,
but over time, you'll

00:02:33.280 --> 00:02:35.710
probably familiarize
yourself with all of these.

00:02:35.710 --> 00:02:36.610
This is oxygen.

00:02:36.610 --> 00:02:38.020
This is nitrogen.

00:02:38.020 --> 00:02:40.140
This is silicon.

00:02:40.140 --> 00:02:42.090
Au is gold.

00:02:42.090 --> 00:02:43.280
This is lead.

00:02:43.280 --> 00:02:49.380
And that most basic unit, of any
of these elements, is the atom.

00:02:52.620 --> 00:02:55.050
So if you were to keep
digging in, and keep

00:02:55.050 --> 00:02:56.940
taking smaller and
smaller chunks of this,

00:02:56.940 --> 00:02:59.739
eventually, you would
get to a carbon atom.

00:02:59.739 --> 00:03:01.530
Do the same thing over
here, eventually you

00:03:01.530 --> 00:03:02.616
would get to a gold atom.

00:03:02.616 --> 00:03:04.490
You did the same thing
over here, eventually,

00:03:04.490 --> 00:03:05.864
you would get
some-- this little,

00:03:05.864 --> 00:03:08.000
small, for lack of a
better word, particle,

00:03:08.000 --> 00:03:09.450
that you would call a lead atom.

00:03:09.450 --> 00:03:11.241
And you wouldn't be
able to break that down

00:03:11.241 --> 00:03:13.560
anymore and still
call that lead,

00:03:13.560 --> 00:03:17.110
for it to still have
the properties of lead.

00:03:17.110 --> 00:03:19.490
And just to give you an idea--
this is really something

00:03:19.490 --> 00:03:21.600
that I have trouble
imagining-- is

00:03:21.600 --> 00:03:24.690
that atoms are
unbelievably small, really

00:03:24.690 --> 00:03:26.220
unimaginably small.

00:03:26.220 --> 00:03:27.750
So for example, carbon.

00:03:27.750 --> 00:03:29.550
My hair is also
made out of carbon.

00:03:29.550 --> 00:03:33.320
In fact, most of me
is made out of carbon.

00:03:33.320 --> 00:03:36.690
In fact, most of all living
things are made out of carbon.

00:03:36.690 --> 00:03:40.840
And so if you took my hair--
and so my hair is carbon,

00:03:40.840 --> 00:03:42.262
my hair is mostly carbon.

00:03:42.262 --> 00:03:43.970
So if you took my
hair-- right over here,

00:03:43.970 --> 00:03:46.075
my hair isn't yellow,
but it contrasts nicely

00:03:46.075 --> 00:03:46.700
with the black.

00:03:46.700 --> 00:03:48.280
My hair is black,
but if I did that,

00:03:48.280 --> 00:03:50.270
you wouldn't be able to
see it on the screen.

00:03:50.270 --> 00:03:51.769
But if you took my
hair, here, and I

00:03:51.769 --> 00:03:55.820
were to ask you, how many
carbon atoms wide is my hair?

00:03:55.820 --> 00:03:57.870
So, if you took a cross
section of my hair, not

00:03:57.870 --> 00:04:00.040
the length, the
width of my hair,

00:04:00.040 --> 00:04:03.384
and said, how many carbon
atoms wide is that?

00:04:03.384 --> 00:04:04.800
And you might
guess, oh, you know,

00:04:04.800 --> 00:04:07.540
Sal already told me
they're very small.

00:04:07.540 --> 00:04:10.016
So maybe there's 1,000 carbon
atoms there, or 10,000,

00:04:10.016 --> 00:04:10.780
or 100,000.

00:04:10.780 --> 00:04:12.000
I would say, no.

00:04:12.000 --> 00:04:14.540
There are 1 million
carbon atoms,

00:04:14.540 --> 00:04:17.579
or you could string 1
million carbon atoms

00:04:17.579 --> 00:04:21.314
across the width of
the average human hair.

00:04:21.314 --> 00:04:22.730
That's obviously
an approximation.

00:04:22.730 --> 00:04:23.970
It's not exactly 1 million.

00:04:23.970 --> 00:04:26.860
But that gives you a sense
of how small an atom is.

00:04:26.860 --> 00:04:28.620
You know, pluck a
hair out of your head,

00:04:28.620 --> 00:04:31.300
and just imagine
putting a million things

00:04:31.300 --> 00:04:34.440
next to each other,
across the hair.

00:04:34.440 --> 00:04:37.450
Not the length of the hair,
the width of the hair.

00:04:37.450 --> 00:04:39.360
It's even hard to see
the width of a hair,

00:04:39.360 --> 00:04:41.110
and there would be a
million carbon atoms,

00:04:41.110 --> 00:04:43.180
just going along it.

00:04:43.180 --> 00:04:46.860
Now it would be pretty
cool, in and of itself,

00:04:46.860 --> 00:04:50.840
that we do know that there
is this most basic building

00:04:50.840 --> 00:04:54.220
block of carbon, this most basic
building block of any element.

00:04:54.220 --> 00:04:57.130
But what's even neater is
that, those basic building

00:04:57.130 --> 00:04:58.570
blocks are related
to each other.

00:04:58.570 --> 00:05:02.400
That a carbon atom is made
up of even more fundamental

00:05:02.400 --> 00:05:02.950
particles.

00:05:02.950 --> 00:05:07.671
A gold atom is made up even
more fundamental particles.

00:05:07.671 --> 00:05:09.170
And depending-- and
they're actually

00:05:09.170 --> 00:05:12.870
defined by the arrangement of
those fundamental particles.

00:05:12.870 --> 00:05:15.430
And if you were to change the
number of fundamental particles

00:05:15.430 --> 00:05:18.430
you have, you could change the
properties of the element, how

00:05:18.430 --> 00:05:22.810
it would react, or you could
even change the element itself.

00:05:22.810 --> 00:05:25.480
And just to understand
it a little bit better,

00:05:25.480 --> 00:05:28.260
let's talk about those
fundamental elements.

00:05:28.260 --> 00:05:30.365
So you have the proton.

00:05:33.020 --> 00:05:36.000
And the proton is actually
the defining-- the number

00:05:36.000 --> 00:05:38.340
of protons in the
nucleus of an atom,

00:05:38.340 --> 00:05:40.610
and I'll talk about the
nucleus in a second-- that

00:05:40.610 --> 00:05:43.010
is what defines the element.

00:05:43.010 --> 00:05:45.702
So this is what
defines an element.

00:05:45.702 --> 00:05:47.660
When you look at the
periodic table right here,

00:05:47.660 --> 00:05:50.490
they're actually written
in order of atomic number.

00:05:50.490 --> 00:05:52.250
And the atomic
number is, literally,

00:05:52.250 --> 00:05:55.140
just the number of
protons in the element.

00:05:55.140 --> 00:05:58.950
So by definition,
hydrogen has one proton,

00:05:58.950 --> 00:06:03.120
helium has two protons,
carbon has six protons.

00:06:03.120 --> 00:06:05.440
You cannot have carbon
with seven protons.

00:06:05.440 --> 00:06:07.150
If you did, it
would be nitrogen.

00:06:07.150 --> 00:06:09.300
It would not be carbon anymore.

00:06:09.300 --> 00:06:10.732
Oxygen has eight protons.

00:06:10.732 --> 00:06:12.940
If, somehow, you were to
add another proton to there,

00:06:12.940 --> 00:06:14.680
it wouldn't be oxygen anymore.

00:06:14.680 --> 00:06:16.370
It would be fluorine.

00:06:16.370 --> 00:06:17.705
So it defines the element.

00:06:20.240 --> 00:06:22.450
And the atomic
number, the number

00:06:22.450 --> 00:06:25.549
of protons-- and
remember, that's

00:06:25.549 --> 00:06:27.340
the number that's
written right at the top,

00:06:27.340 --> 00:06:30.260
here, for each of these
elements in the periodic table--

00:06:30.260 --> 00:06:33.010
the number of protons is
equal to the atomic number.

00:06:36.705 --> 00:06:38.080
And they put that
number up here,

00:06:38.080 --> 00:06:42.160
because that is the defining
characteristic of an element.

00:06:42.160 --> 00:06:46.050
The other two constituents
of an atom-- I

00:06:46.050 --> 00:06:47.430
guess we could
call it that way--

00:06:47.430 --> 00:06:51.780
are the electron
and the neutron.

00:06:55.390 --> 00:06:57.110
And the model you
can start to build

00:06:57.110 --> 00:07:01.320
in your head-- and this model,
as we go through chemistry,

00:07:01.320 --> 00:07:03.690
it'll get a little bit more
abstract and really hard

00:07:03.690 --> 00:07:04.976
to conceptualize.

00:07:04.976 --> 00:07:06.350
But one way to
think about it is,

00:07:06.350 --> 00:07:08.430
you have the protons
and the neutrons that

00:07:08.430 --> 00:07:09.850
are at the center of the atom.

00:07:09.850 --> 00:07:11.710
They're the nucleus of the atom.

00:07:11.710 --> 00:07:15.060
So for example, carbon,
we know, has six protons.

00:07:15.060 --> 00:07:19.660
So one, two, three,
four, five, six.

00:07:19.660 --> 00:07:22.420
Carbon-12, which is
a version of carbon,

00:07:22.420 --> 00:07:24.110
will also have six neutrons.

00:07:24.110 --> 00:07:26.100
You can have versions
of carbon that

00:07:26.100 --> 00:07:27.940
have a different
number of neutrons.

00:07:27.940 --> 00:07:30.330
So the neutrons can change,
the electrons can change,

00:07:30.330 --> 00:07:31.830
you can still have
the same element.

00:07:31.830 --> 00:07:33.280
The protons can't change.

00:07:33.280 --> 00:07:36.480
You change the protons, you've
got a different element.

00:07:36.480 --> 00:07:41.300
So let me draw a carbon-12
nucleus, one, two, three, four,

00:07:41.300 --> 00:07:43.300
five, six.

00:07:43.300 --> 00:07:46.450
So this right here is
the nucleus of carbon-12.

00:07:46.450 --> 00:07:48.910
And sometimes, it'll
be written like this.

00:07:48.910 --> 00:07:52.720
And sometimes, they'll actually
write the number of protons,

00:07:52.720 --> 00:07:53.840
as well.

00:07:53.840 --> 00:07:56.510
And the reason why we
write it carbon-12--

00:07:56.510 --> 00:07:58.830
you know, I counted
out six neutrons--

00:07:58.830 --> 00:08:00.760
is that, this is
the total, you could

00:08:00.760 --> 00:08:04.206
view this as the total number
of-- one way to view it.

00:08:04.206 --> 00:08:05.580
And we'll get a
little bit nuance

00:08:05.580 --> 00:08:08.370
in the future-- is that this
is the total number of protons

00:08:08.370 --> 00:08:11.870
and neutrons inside
of its nucleus.

00:08:11.870 --> 00:08:15.172
And this carbon, by definition,
has an atomic number of six,

00:08:15.172 --> 00:08:16.630
but we can rewrite
it here, just so

00:08:16.630 --> 00:08:18.870
that we can remind ourselves.

00:08:18.870 --> 00:08:21.550
So at the center of a carbon
atom, we have this nucleus.

00:08:21.550 --> 00:08:25.000
And carbon-12 will have six
protons and six neutrons.

00:08:25.000 --> 00:08:26.970
Another version of
carbon, carbon-14,

00:08:26.970 --> 00:08:29.040
will still have six
protons, but then it

00:08:29.040 --> 00:08:31.018
would have eight neutrons.

00:08:31.018 --> 00:08:32.559
So the number of
neutrons can change.

00:08:32.559 --> 00:08:34.950
But this is carbon-12,
right over here.

00:08:34.950 --> 00:08:38.990
And if carbon-12 is neutral--
and I'll give a little nuance

00:08:38.990 --> 00:08:41.419
on this word in a second
as well-- if it is neutral,

00:08:41.419 --> 00:08:43.890
it'll also have six electrons.

00:08:43.890 --> 00:08:47.690
So let me draw those six
electrons, one, two, three,

00:08:47.690 --> 00:08:49.910
four, five, six.

00:08:49.910 --> 00:08:52.190
And one way-- and this is
maybe the first-order way

00:08:52.190 --> 00:08:54.980
of thinking about
the relationship

00:08:54.980 --> 00:08:57.380
between the electrons
and the nucleus--

00:08:57.380 --> 00:09:00.340
is that you can imagine the
electrons are, kind of, moving

00:09:00.340 --> 00:09:02.930
around, buzzing
around this nucleus.

00:09:02.930 --> 00:09:04.540
One model is, you
could, kind of,

00:09:04.540 --> 00:09:06.790
thinking of them as
orbiting around the nucleus.

00:09:06.790 --> 00:09:08.050
But that's not quite right.

00:09:08.050 --> 00:09:10.460
They don't orbit the
way that a planet, say,

00:09:10.460 --> 00:09:11.710
orbits around the sun.

00:09:11.710 --> 00:09:13.900
But that's a good
starting point.

00:09:13.900 --> 00:09:16.430
Another way is, they're kind
of jumping around the nucleus,

00:09:16.430 --> 00:09:18.680
or they're buzzing
around the nucleus.

00:09:18.680 --> 00:09:20.680
And that's just
because reality just

00:09:20.680 --> 00:09:22.100
gets very strange at this level.

00:09:22.100 --> 00:09:24.183
And we'll actually have
to go into quantum physics

00:09:24.183 --> 00:09:26.570
to really understand what
the electron is doing.

00:09:26.570 --> 00:09:29.320
But a first mental
model in your head

00:09:29.320 --> 00:09:32.740
is at the center of this
atom, this carbon-12 atom,

00:09:32.740 --> 00:09:37.000
you have this nucleus,
right over there.

00:09:37.000 --> 00:09:40.700
And these electrons are
jumping around this nucleus.

00:09:40.700 --> 00:09:43.850
And the reason why these
electrons don't just

00:09:43.850 --> 00:09:45.360
go off, away from this nucleus.

00:09:45.360 --> 00:09:47.570
Why they're kind of
bound to this nucleus,

00:09:47.570 --> 00:09:49.860
and they form part
of this atom, is

00:09:49.860 --> 00:09:54.570
that protons have
a positive charge

00:09:54.570 --> 00:09:57.990
and electrons have
a negative charge.

00:09:57.990 --> 00:10:01.874
And it's one of these properties
of these fundamental particles.

00:10:01.874 --> 00:10:03.290
And when you start
thinking about,

00:10:03.290 --> 00:10:04.690
well, what is a
charge, fundamentally,

00:10:04.690 --> 00:10:05.481
other than a label?

00:10:05.481 --> 00:10:07.020
And it starts to
get kind of deep.

00:10:07.020 --> 00:10:08.810
But the one thing
that we know, when

00:10:08.810 --> 00:10:10.670
we talk about
electromagnetic force,

00:10:10.670 --> 00:10:13.280
is that unlike charges
attract each other.

00:10:13.280 --> 00:10:14.890
So the best way
to think about it

00:10:14.890 --> 00:10:17.100
is, protons and
electrons, because they

00:10:17.100 --> 00:10:19.990
have different charges,
they attract each other.

00:10:19.990 --> 00:10:21.520
Neutrons are neutral.

00:10:21.520 --> 00:10:25.070
So they're really just sitting
here inside of the nucleus.

00:10:25.070 --> 00:10:30.700
And they do affect the
properties, on some level,

00:10:30.700 --> 00:10:33.330
for some atoms of
certain elements.

00:10:33.330 --> 00:10:36.340
But the reason why we have the
electrons not just flying off

00:10:36.340 --> 00:10:37.970
on their own is
because, they are

00:10:37.970 --> 00:10:42.470
attracted towards the nucleus.

00:10:42.470 --> 00:10:45.690
And they also have an
unbelievably high velocity.

00:10:45.690 --> 00:10:48.630
It's actually hard for-- and
we start touching, once again,

00:10:48.630 --> 00:10:51.260
on a very strange
part of physics

00:10:51.260 --> 00:10:54.380
once we start talking about what
an electron actually is doing.

00:10:54.380 --> 00:10:56.780
But it has enough, I
guess you could say,

00:10:56.780 --> 00:10:59.690
it's jumping around enough that
it doesn't want to just fall

00:10:59.690 --> 00:11:04.520
into the nucleus, I guess is
one way of thinking about it.

00:11:04.520 --> 00:11:08.220
And so I mentioned,
carbon-12 right over here,

00:11:08.220 --> 00:11:09.820
defined by the
number of protons.

00:11:09.820 --> 00:11:12.710
Oxygen would be defined
by having eight protons.

00:11:12.710 --> 00:11:16.420
But once again, electrons can
interact with other electrons.

00:11:16.420 --> 00:11:19.140
Or they can be taken
away by other atoms.

00:11:19.140 --> 00:11:23.380
And that actually forms a lot of
our understanding of chemistry.

00:11:23.380 --> 00:11:26.400
It's based on how many
electrons an atom has,

00:11:26.400 --> 00:11:27.860
or a certain element has.

00:11:27.860 --> 00:11:29.580
And how those electrons
are configured.

00:11:29.580 --> 00:11:33.990
And how the electrons of
other elements are configured.

00:11:33.990 --> 00:11:36.490
Or maybe, other atoms
of that same element.

00:11:36.490 --> 00:11:41.180
We can start to predict
how an atom of one element

00:11:41.180 --> 00:11:43.360
could react with another
atom of that same element.

00:11:43.360 --> 00:11:46.700
Or an atom of one element,
how it could react,

00:11:46.700 --> 00:11:48.780
or how it could
bond, or not bond,

00:11:48.780 --> 00:11:51.370
or be attracted, or
repel, another atom

00:11:51.370 --> 00:11:52.630
of another element.

00:11:52.630 --> 00:11:55.230
So for example-- and we'll
learn a lot more about this

00:11:55.230 --> 00:12:00.110
in the future-- it is possible
for another atom, someplace,

00:12:00.110 --> 00:12:03.180
to swipe away an
electron from a carbon,

00:12:03.180 --> 00:12:05.840
just because, for
whatever reason.

00:12:05.840 --> 00:12:09.780
And we'll talk about certain
elements, certain neutral atoms

00:12:09.780 --> 00:12:12.360
of certain elements,
have a larger

00:12:12.360 --> 00:12:14.010
affinity for
electrons than others.

00:12:14.010 --> 00:12:17.140
So maybe one of those swipes
an electron away from a carbon,

00:12:17.140 --> 00:12:19.540
and then this carbon
will be having

00:12:19.540 --> 00:12:21.730
less electrons than protons.

00:12:21.730 --> 00:12:25.370
So then it would have five
electrons and six protons.

00:12:25.370 --> 00:12:28.152
And then it would have
a net positive charge.

00:12:28.152 --> 00:12:30.110
So, in this carbon-12,
the first version I did,

00:12:30.110 --> 00:12:33.170
I had six protons,
six electrons.

00:12:33.170 --> 00:12:34.530
The charges canceled out.

00:12:34.530 --> 00:12:37.170
If I lose an electron, then
I only have five of these.

00:12:37.170 --> 00:12:39.210
And then I would have
a net positive charge.

00:12:39.210 --> 00:12:41.241
And we're going to talk
a lot more about all

00:12:41.241 --> 00:12:42.990
of this throughout the
chemistry playlist.

00:12:42.990 --> 00:12:44.200
But hopefully, you
have an appreciation

00:12:44.200 --> 00:12:46.241
that this is already
starting to get really cool.

00:12:46.241 --> 00:12:48.300
Once we can already
get to this really,

00:12:48.300 --> 00:12:52.937
fundamental building
block, called the atom.

00:12:52.937 --> 00:12:55.520
And what's even neater is that
this fundamental building block

00:12:55.520 --> 00:12:58.800
is built of even more
fundamental building blocks.

00:12:58.800 --> 00:13:00.630
And these things
can all be swapped

00:13:00.630 --> 00:13:03.160
around, to change the
properties of an atom,

00:13:03.160 --> 00:13:05.880
or to even go from an
atom of one element

00:13:05.880 --> 00:13:08.466
to an atom of another element.