In the last video we defined our random variable x as the
number of heads we get after flipping a coin five times,
and it's a fair coin.
And then we figured out the different probabilities that
the random variable could take on different values.
Let me rewrite them all here.
The probability that we get exactly one head or
zero heads-- actually, let's start there.
The probability we get zero heads, and that's the same
thing as getting five tails-- we figured out was 1 out of 32.
The probability of getting 1 head we figured
out was 5 over 32.
The probability of getting 2 heads-- probability that our
random variable is equal to 2, what was that?
I think it was 10/32.
Yeah, it was 10/32 or 5/16.
10/32 which is equal to 5/16.
The probability that we get three heads, that was actually
equal to the same thing, 10/32 which is equal to 5/16 .
And that made sense because the probability of getting 3
heads is the same thing as a probability of getting 2 tails.
And the probability of getting 2 tails should really be the
same thing as the probability of getting 2 heads.
And then finally, or almost finally, the probability that
we get 4 heads, that was equal to 5/32, which also makes sense
because the probability of getting 4 heads is the same
thing as the probability of getting 1 tail, right?
And 1 tail should be the same thing as 1 head, so that's why
these numbers are the same.
And then finally, the probability that we get all 5
heads, which is the same thing as getting no tails, is 1/32.
The probability of no tails is the same thing as the
probability as no heads.
So let's draw this.
Let's draw our probability distribution.
OK, let me draw the x-axis.
That's not thick enough.
I'll do it a different color.
All right, and just let me try draw the y-axis.
Almost there.
All right, so what are the different values that we care
about for our random variable?
Notice, these are all discreet values.
They're all particular finite numbers that we care that our
random variable can take on.
And that was actually a function of our definition of
the random variable, right?
This random variable, number of heads after 5 flips-- you can't
have an infinite number of values here.
Your value can either be 0 through 5, and it can't
be a non-integer even.
But the important thing to realize that there's a finite
number of values that this random variable can take on and
that's why we have a discreet probably distribution.
All right, so the probability that it takes on-- let me just
draw 0, 1, 2, 3, 4, and 5.
This is for our random variable, x.
So this is the number of heads in 5 trials.
Let me draw the high points first.
So let's see, the high points are-- they're all right there.
10/32.
That's the probability of 2 and 3 and they're the same.
Let me draw little bar graph.
So 2 and 3; let's just say that this is 10/32 right there.
I'll draw them the highest because those are
the highest values.
2 and 3 both hit 10/32.
Actually, let me draw it a little differently.
Let me make it so they touch each other.
So let's say 2 is there, and then we draw 3.
And 3 is also there.
There you go.
All right, what's the probability that you get 1?
That was 5/32, so that's 1/2 of these.
These two should be the same height.
So that's 5/32.
Should be like that.
That's the same thing as getting 4 heads,
so that's like that.
Let's assume that this 4 is down here.
The probability of getting either 0 or
5 is the same thing.
That's 1.
So that's going to be 1/5 of this height.
So it's going to look something like that.
Let's say this last one is for 5.
The drawing is the hardest part.
That's going to look something like that.
All right, and so this is for 5 and this is for 4 right here.
And this right here is 10/32 or 5/16.
This is 5/32.
And this right here is 1/32.
What I have just drawn is a binomial probability
distribution.
It's a particular instance of the binomial
probability distribution.
And actually, as you get to kind of an infinite number of
values, and once we get to the continuous, this'll approach
the famous bell curve that you hear about where
things actually start looking like a bell.
Or it'll start looking like that.
And we'll do some-- maybe I'll get Excel out and I'll do it
where I start saying-- instead of saying, the number
of heads in 5 trials.
If I did this as the number of heads in 5 million trials,
these bars would get very, very close together and you'd start
seeing them approach this thing that looks like a bell curve.
And this is a very important distribution.
One, because it actually describes a lot of
random processes.
But it's actually very important to statistics
generally because a lot of times in statistics you don't
know the underlying mechanism that's generating your results
and you just assume that there's just a bunch of
random stuff going on.
You just add up a bunch of random events.
When you're adding up random events that's like counting
the number of heads.
So you assume that a lot of things have-- well,
in the discreet case, a binomial distribution.
And then, in a few videos from now I'll show you the
continuous cases, which is the normal distribution.
And this is really important to realize because sometimes
people make that assumption about certain things and the
biggest one-- and this is probably the most negative
repercussions, especially in this type of financial
environment.
Well, they'll assume that some process has a binomial or a
normal distribution where it really doesn't.
Because if you assume that something has a distribution
like this you start saying, oh, well the probabilities at
these ends are really low.
But what if the distribution is something like this?
And I'll do more work on this.
I don't want to get too advanced too fast.
But the general idea, it's really important to understand
the assumption you're making when someone says, oh, we
assume it's a normal distribution.
We assume it's a binomial distribution.
But with that said, that's a little bit of a nugget
of why this is important going into the future.
But just to kind of review a little more what we've
already studied, let's think about this.
Why is this called a binomial distribution?
Well, if you think about it, when you flipped 5 coins or
when you had 5 flips of the same coin-- each arrangement,
I'll pick an arbitrary arrangement of heads and tails.
Each circumstance of heads and tails, I guess I could call it.
You know, that's one of them, 5 flips.
Each of these have exactly a 1 in 32 probability.
In fact, there's exactly 32 ways I can draw some
combination of heads and tails here.
So each of those have a 1 in 32 probability.
And what we essentially did in the last video when we said,
OK, what was the probability of getting exactly two heads?
We essentially said, OK, how many of these circumstances, or
how many of these-- I guess we could call it-- how many of
these permutations had 2 heads?
And we counted them up and we said, 10 of them have 2 heads
and that's why the probability was 10 out of 32.
And the way we figured that out is for example,
if we said, 2 heads.
We said, OK, 1 of 5 flips could be the first head, and then 1
of 4 flips could be the second head.
And then we don't care about order, so we don't care if flip
1 was a first head and flip two was a second head or was
the other way around.
So we divide by 2, and we said that this was the
same thing as 5 factorial.
Actually, let me do it color coded.
This part is the same thing as 5 factorial over 3 factorial.
And then, that's actually the same thing as 2 factorial
because 2 factorial is 2 times 1.
So in general, let me write this down.
The probability that x equaled n-- and this is
for the x that I defined.
The random variable, the number of heads I get in 5 flips of
the coin-- it was equal to 5 factorial divided
by n factorial.
In this case it was 2, 2 was n.
We said was the probability that we get exactly 2 heads.
Times 5 minus 2 factorial.
And I encourage you to review because it really is important
to get an intuitive feel of this.
I encourage you to review the videos I made in the
probability playlist on binomial coefficients, and on
this flipping of coins type of probability problems because I
go a little more in detail in the actual intuition of this.
But if you watch those or if you have a little bit of
experience with this, you would recognize that this is
the binomial coefficient.
And actually, I even have a video why it's called the
binomial because these coefficients show up when you
actually multiply binomials.
Binomials are just things like if I just have x plus why and I
start multiplying it by itself, take it to different powers,
the coefficients of these tend to be-- well, they are
the same as these.
And I give you a whole video on why that-- well,
that's why that works.
And so this is why it's even called the
binomial coefficient.
And why this is called the binomial distribution.
But another way of writing this, the shorthand
is 5 choose n.
And they say that because what were we doing?
We said we had 5 flips and we were choosing 2 of
the flips to be heads.
Or, in this case, we have 5 flips and we're choosing n
of the flips to be heads.
So this will tell you how many of the different permutations
satisfy our condition.
And then we say, well, what's the probability
of each of those?
It's times 1 out of 32.
So this is why it's called a binomial distribution.
Each of the values of probabilities for each of the
random variable values-- you can figure them out by using
your binomial coefficients.
Anyway, I just realized I'm out of time again.
And don't worry, I'm going to do a couple of more examples
with this because I really want you to get the hang of the
binomial distribution before we move into the normal
distribution.
See you soon.