We now know what a
probability distribution is. It could be a discrete
probability distribution or a continuous one, and we learned
that that's a probability density function. Now let's study a couple
of the more common ones. So let's say I have a coin,
and it's a fair coin, and I'm going to flip it five times. And I'm going to define
my random variable, X, I'll define it. I'll make it a capital X. It equals the number of
heads I get after 5 flips. Maybe I flip them all at once. Maybe I have 5 coins and I
flip them all at once and I just count the heads. Or I could have one coin and I
could flip it five times and see the number of heads. It actually doesn't matter. But let's say I have 1 coin and
I flip it five times, just so we have no ambiguity. So this is my definition
of my random variable. As we know, a random variable,
it's a little different than a regular variable, it's
more of a function. It assigns a number with
an experiment and this one's pretty easy. We just count the number of
heads we got after 5 flips and that's our random variable, X. Let's think a little bit of
what are the different probabilities of getting
different numbers here? So what is the probability
that X, big capital X, is equal to 0? So what's the probability that
you get no heads after 5 flips? Well, that's essentially the
same thing as the probability of getting all tails. This is a bit of a
review of probability. You have to get all tails. And what's the probably
of each of these tails? Well, it's 1/2. So it'd have to be 1/2
times 1/2 times 1/2 times 1/2 times 1/2. So it'd have to be 1/2
to the fifth power. 1 to the fifth is 1 over
2 to the fifth is 32. Fair enough. Now what's the probability--
I'm going through a little bit of a probability review. Just because I think it's
important just to get the intuition of where we're going
now and how you actually form a discrete probably distribution. Now what's the probability
that you get exactly 1 head? Well, that could just
be the first head. It could be heads, then tails,
tails, tails, tails, tails. Or it could be the second head. It could be the probability of
tails, heads, tail, tails, tails and so forth. This one head that you get, it
could be in any of the 5 spots. So what's the probability of
each of these situations? Well the probability that
you get a head is 1/2. Then the probability you
get tails is 1/2 times 1/2 times 1/2 times 1/2. So the probability of each of
these situations is 1/32. Just like the probability of
this particular situation. In fact, the probability of any
particular order of heads and tails is going to
be 1 out of 32. There's actually 32
possible scenarios. So the probability of
this is 1 out of 32. The probability of
this is 1 out of 32. And there's situations like
this because the heads could be in any of the 5 spots. So the probability that we have
exactly one head is equal to 5 times 1/32, which
is equal to 5/32. Fair enough. Now it gets interesting. What is the probability--
I'll do each of these in a different color. What is the probability that my
random variable is equal to 2? So I flip the coin five times,
what is the probability that I get exactly 2 heads? Now it becomes a little
bit interesting. So what are all the situations? I could have heads, heads,
tails, tails, tails. I could have a head, tails,
heads, tails, tails. And if you think about it
there's these two heads and they can go in a bunch of
different places, and it starts to get a little bit confusing. You can't just think of it
in kind of the scenario analysis like we did here. You can, but it becomes
a little bit confusing. You have to realize one thing. Each of the scenarios, there's
a 1 out of 32 probability. 1/2 times 1/2 times 1/2
times 1/2 times 1/2. So that's a 1 out of 32
probability-- each of those. And we have to think about, how
many of these scenarios satisfy our condition-- 2 heads? So essentially you could
imagine we have 5 flips and we're going to choose
2 of them to be heads. So you could almost imagine if
you had all the flips sitting around, and we had two chairs,
and we said, OK, whichever flips sit in these chairs, they
get to be the heads chair. Or they get to be
the heads flips. And we don't care of
what order they sit in. I'm going some place with this,
just so you get hopefully, a little intuition. And you might want to watch
some of the probability videos on this when I talk about the
binomial theorem and all that. Because I go into this in
a little bit more detail. But if you think about it that
way, the binomial coefficient kind of starts to make sense. Because if you think about,
OK, I have 5 heads-- I have five flips, sorry. Which flip is going to
be the first heads? Well, there's 5 possibilities. Let me do this in a
different color. There's 5 possibilities for
which of the positions or which of the flips is going
to be the first head. Now how many possibilities are
there for the second head? Well, the first flip that
we used, used up one of the heads chairs. Or sorry, the first heads
chair, the first heads spot was used up by one of the flips. Now there's only 4 flips left,
so there's only 1 out of 4 flips that the second
head could be in. And you saw that here. I picked the first one to be a
heads here and then we said, OK, 1 of these 4 also
have to be a head. Or if I said, OK, this is the
first head, then either this one, this one, this one, or
this one has to be heads. So there's only 4
possibilities. So all I'm saying is the first
time around you have 5 different possibilities for
where the first heads could be. And then the second
time around you have 4 different possibilities. You have to think about it. When we count it just like this
we're being dependent on order. But we don't care which
flip is in which heads. We're not saying that this is a
heads 1 or this is a heads 2. These are both heads. It doesn't matter. We could have this being
the head seat 1, this could be head seat 2. Or, it could be the
other way around. This could be the second head
spot, and this could be the first head spot. And I'm saying that just
because it's important to realize the distribution of
permutation and a combination. We don't care about order. So there's actually two
different ways that this can happen. So we divide it by 2. And as you'll see it's
actually 2 factorial ways that it can happen. If this had 3 we would do 3
factorial, and I'll show you how that can happen. And so this will be equal to--
5 times 4 is 20, divided by 2, which is equal to 10. So there's 10 different
combinations out of the 32 where you have exactly 2 heads. So 10 times 1 out of 32
equal to 10/32, which is equal to what? 5/16. And actually, let me
write this in terms of a binomial coefficient. This up here, that number right
there, if you think about it-- that's the same thing as
5 factorial over, what's 5 times 4? 5 factorial is 5 times 4
times 3 times 2 times 1. So if I just want 5 times 4
what I can do is I divide 5 factorial divided
by 3 factorial. This is equal to 5 times 4
times 3 times 2 times 1 divided by 3 times 2 times 1. And you're just left
with the 5 times 4. So this is the same
thing as that. And then, since we didn't
care about order, we wanted the 2 here. And actually, it turns out
that it's 2 factorial. I'll show you that
in a little bit. It's two factorial times 1/32. This was the probability that
our random variable-- that we have exactly 2 heads. Now what's the probability
that we have exactly 3 heads? The probability that
x is equal to 3. So by the same logic, the
first head spot can be taken by 1 of the 5 flips. Then the second head spot
can be taken by 1 of the 4 left remaining flips. And then the third head spot
could be taken by 1 of the 3 remaining flips. And then, how many different
ways can I arrange 3 flips? In general, how many ways
can you arrange 3 things? And it's 3 factorial. And you could work that out or
you might want to watch the probability videos where I work
that a little bit better. But actually, If if you just
take the letters A, B, and C there's 6 ways that you
can arrange these. You can view these
as the head spots. And we don't care about order,
so it could be ACB, CAB. It could be BAC, BCA, and
then what's the last one that I haven't done? It's CBA. There are 6 ways to arrange
3 distinct things. We're dividing by it because we
don't want to double count these 6 different ways because
we're viewing them all as the same thing. Not in this case, but in the
case of we don't care which flip is sitting in
which head spot. So that's why I got
the 3 factorial. And this is the same thing. 5 times 4 times 3-- this could
be written as 5 factorial divided by 3 factorial. Then this is divided
by 3 factorial. This one is this one. And so this is equal to-- I
don't know-- 3 factorial is equal to 3 times 2 times 1. The 3's cancel out. This becomes a 2. This becomes a 1. Once again, 5 times
2, so it's 10. Each situation has a 1 in
32 probability, so once again it's equal to 5/16. And that's interesting. The probability that you get
3 heads is the same as the probability you get 2 heads. And the reason that-- well,
there's a lot of reasons why that's the case. But if you think about the
probability of getting 3 heads is the same thing as the
probability of getting 2 tails. And the probability of getting
2 tails should be the same thing as the probability
of getting 2 heads. And so it's nice that the
numbers work out that way. All right, so we're
almost there. What's the probability
that x is equal to 4? Well, we could use the same
kind of formula now that we were using before. You know, it could be 5
times 4 times 3 times 2. So that's 5 times 4 times 3
times 2-- all the way, how many ways can you arrange 4 things? It's 4 factorial. 4 factorial is essentially
this thing right here. 4 times 3 times 2 times-- this
is 4 times 3 times 2 times 1. So these cancel out, so it's 5
and then each of the scenarios has a 1 in 32 chance. So it's equal to 5/32. And once again, notice, the
probability of getting 4 heads is the same thing as the
probability of getting exactly 1 head. And that makes sense because
4 heads is the same thing as getting exactly 1 tail. And you say, oh, where's
that one tail going to be showing up? Oh, there's 5 different
spot for it. And each of the scenarios
has a 1 in 32 possibility. And then finally, what's
the probability that x is equal to 5? You get all 5 heads. You know, it's going
to be heads, heads, heads, heads, heads. Each of these have
a 1/2 probability. You multiply them,
you get 1 out of 32. Or another way to think about
it, if you think about it, the 32 different ways that you can
have heads and tails in these experiments-- this is only
one of those circumstances. This was 5 of the
circumstances. This was 10 of the
circumstances. Anyway, we've done the work,
now we're ready to draw a probability distribution. I'm running out of time. Actually, let me continue
that in the next video. And if you're in the mood,
maybe you draw it before you watch the next video. See you soon.
