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.