Let's explore another example of a binomial distribution.
Let's say I'm playing basketball and I know I'm going
to take 10 shots in the game.
So I'm going to take 10 shots.
Let's say that n is equal to 10.
That's the number of shots I take.
Let's just say that every shot I take they're independent
events and that, in general, I have a 30% chance of
making any given basket.
We're not going to get too detailed of where I take
the basket from or are they three pointers?
Or are they lay out?
Whatever.
Every time I take a shot I have a 30% chance of making it.
So let's say the probability of success is equal to 30%.
Probability is equal to 30%.
I just made that definition.
Let's define a random variable, X, like we always do.
X is equal to number of shots I make.
Number of shots/baskets I make.
And this one's a little bit more interesting than the
flipping a coin example because in the flipping a coin example,
heads or tails both had an equal probability of happening.
In this situation, making a shot is less likely
than not making a shot.
The not making a shot-- let's say not.
Not making a shot, that is equal to 1 minus the
probability of making a shot, which is equal to 70%.
So there's a slight twist on what we did before, but in some
ways this might be a more interesting example to
your everyday life.
And let's see what the distribution looks like.
And also, I'm taking more shots now than I did before.
So it'll probably take a little bit more time-- actually,
this is too much.
I don't want to waste too much time.
Let's say I take 6 shots.
And the random variable, X, is the number of shots I make.
So let's think about how we figure out this probability.
What is the probability that X is equal to 0?
So I make no shots at all.
So I take 6 shots and every shot I take I miss.
So in order for that to happen this event has to happen
six times in a row.
So something with a 70% probability has to happen six
times in a row, the probability of this is 0.7-- it's 705--
times 0.7 times 0.7 times 0.7-- I'm getting confused.
It's 0.7 times 0.7 times 0.7 times 0.7 times 0.7 times 0.7.
Each of these are missing the first shot, second shot, third
shot, and so forth and so on.
And this is equal to 0.7 to the sixth power.
Whatever that might be.
All right, and there's only one way to do it.
I literally have to miss every shot in order for
X to be equal to 0.
What's the probability that I make exactly 1 shot?
I might make the first one and then miss the other ones.
I might miss all of them but the second.
I might miss all of them, but the third.
So let's think about, what's the probability of each
of those circumstances?
Let's say I make it.
I'll call that a make.
Too bad make and miss both start with m.
Let's see, I'll call it point-- point and miss.
So it could be like a point and then 5 misses.
it?
Could be a miss, a miss, a miss, a point, a miss, a miss.
Now you can imagine, there's 5 of these scenarios.
There's actually 6 of these scenarios.
The shot that I make would be in 1 of these 6-- I don't want
to use the word bucket because we're already using a
basketball analogy.
But it can be in 1 of these 6 spots.
In one of the six shots that I take.
So there's 6 of these scenarios.
What's the probability of each of these?
This is a 30% chance of happening.
And then each of these are 70% chance of happening.
So it'd be 0.3 times 0.7 times 0.7 times 0.7
times 0.7 times 0.7.
This is 0.7 to the fifth.
Times 0.7 to the fifth.
That's the odds of this happening.
What are the odds of this one happening?
Well, let's see.
You have 0.7 times 0.7 times 0.7 times 0.3
times 0.7 times 0.7.
But if you think about it, you're still taking 0.7
times itself five times.
0.7, 0.7, 0.7, 0.7, 0.7.
You're taking 0.7 to the fifth power and you have to multiply
by 0.3 once as well.
So no matter where you make the shot, the chances of any of
these permutations independently are 0.3 times
0.7 to the fifth-- whatever that is.
And then how many ways are there to do this?
Well, we just figured out.
There are 6 ways to do this.
You might make only the first shot.
You might make only the second shot, only the third shot,
and so forth and so on.
So the probability that X is equal to 1-- the probability
that our random variable is equal to 1 is equal to 6 times
0.3 times 0.7 to the fifth.
And just so that we make it clear and connect it all to the
binomial distribution, if we were to have done the n
choose 0 here, what's the binomial coefficient?
So in our example n is 6, so what's 6 choose 0?
That's 6 factorial over 0 factorial, times 6
minus 0 factorial.
6 minus 0 is just 6.
So 6 factorial divided by 6 factorial, those cancel out.
You're left with 1.
But what's 0 factorial?
And this is one of those bizarre definitional things in
mathematics, and I'll leave you to think about it.
And I've addressed this is previous videos.
But 0 factorial is actually-- so that it works out properly
is defined to be equal to 1.
And I did that to just show you that this is the binomial
coefficient on this term.
So we just multiply it by 1.
That's why I never even brought it up.
The probability of this happening is 0.7 to the sixth.
And then you multiply it times the binomial coefficient.
But there's only one way that this can happen and that's
why this turned out to be 1.
I didn't want to confuse you so I didn't bring all that up.
But we did still use the binomial coefficient.
Let's think about it in this one.
In this situation we're taking 6 shots, and we're choosing
only 1 of them to be made.
What's 6 choose 1?
That's 6 factorial over 1 factorial, divided by
6 minus 1 factorial.
That's 6 factorial divided by-- well, 1 factorial
can be ignored.
That's just one.
Divided by 5 factorial.
Well, what's that?
That's 6 times 5 times 4 times 3 times 1 divided by
5 times 4 times 3 times 1.
So everything else cancels out except for just
four of the 6's.
And so that's where we got our 6 from.
We got it by reasoning, which actually I think is
a better way to get it.
But I just wanted to show you that we're still using the
binomial coefficients.
This is 6 choose 1.
And then we multiply that times the probability of any of
these permutations happening.
And we figure that out by we make 1 shot and
we miss the rest.
Let's keep going.
I think you'll get the hang on this sooner than later.
What's the probability that you make exactly two shots?
So what's the probability for any given-- let's say I miss,
miss, miss, miss, and then I get 2 points.
Or two shots.
I don't want to get too confused.
Let's say they're all worth 1 point in this version of
basketball we're playing.
So here, what's the probability?
If 0.7 times 0.7 times 0.7 times 0.7 times 0.3 times 0.3.
So this is 0.7 to the-- 1, 2, 3, fourth power--
times 0.3 squared.
That's each of these circumstances.
But this isn't the only way that I can make 2 shots.
essentially I can choose any two of these shots I take to
be the ones that I make.
I'm not picking it, but the god's of probability
will pick it.
So this isn't the only circumstance.
The probability of just this circumstance, where I miss
exactly the last 2 shots is this.
But if I wanted to figure out all of the different ways that
I can make exactly 2 shots, I would essentially say, well,
I'm taking 6 shots and I'm choosing 2 of them to be made.
So how many does that result in?
Let's see.
6 factorial over 2 factorial 6 minus 2 factorial.
That is equal to-- I like to multiply it out.
That's 6 times 5 times 4 times 2 times 1 over 2 times 1.
This is 4 factorial, 6 minus 2 is 4 factorial.
4 times 3 times 2 times 1.
And actually, I forgot to write a 4 up here.
6 times 5 times 3 times 3 times 2 times 1.
Anyway, this cancels out with that.
The 2 and the 6 is a 3, so it becomes 15.
So there are 15 possible ways to make exactly two shots,
especially if you don't care about the order in which-- I'm
not saying that this point and this point, if they were
to happen the other way around, it doesn't matter.
It's kind of the same circumstance.
I made the last two shots.
It doesn't say I made the second to last shot in one
way and I made the last shot in the other way.
We're not differentiating between, we're just saying
that we made them.
So that's why there's 15 different ways to make
2 shots out of 6.
And the probability of each of those is 0.7 to the
fourth times 0.3 squared.
So the probability of making exactly 2 shots is going to be
6 choose 2 times 0.7 to the fourth times 0.3 squared.
And we can go on.
Let's do them fast.
The probability that I make exactly 3 shots by the same
logic-- well, what's the probability that I make
exactly-- in any one of the circumstance-- well, how many
ways can I make 3 shots?
Well, I'm taking 6 and I'm choosing 3.
And then the probability of each of those ways is, in
order to make 3 shots I'm going to miss 3 shots.
And then I'm going to make 3 shots.
That's straightforward enough and we could calculate
what that is.
But hopefully you know it.
Let me just do it.
So that's 6 factorial over 3 factorial times 6 minus 3
factorial times this part-- 0.7 third time's 0.3 squared.
Let's keep going and this should actually get faster once
you-- see probability that I make exactly 4 shots.
Well, I'm taking 6 and I'm going to make 4 of them.
So I have to choose 4 out of 6.
So if I'm making 4 shots I missing 2 shots, so
there's going to be 2 shots that I miss.
0.7 times 0.7.
That's the probability of a miss-- 0.7.
And then I making 4 of them.
So 0.3-- there's a 30% chance of making each of them.
So to make 4 it's 0.3 to the fourth.
So in any one of these ways where I make 4 shots, this
is their probability.
And there are this many ways of doing it.
And that's equal to 6 factorial over 4 factorial times
6 minus 4 factorial.
And then times 0.7 squared times 0.3 to the fourth.
And if you think, this is 2 factorial.
We figured out what that was up here.
6 choose 2 was the same thing.
It was 6 factorial over 2 factorial divided
by 4 factorial.
It's the same thing as this, we just switched
the 4's and the 2's.
So this would also be equal to 15.
Anyway, I'll probably do that in the next video,
but let's just calculate these really fast.
The probability that X is equal to 5?
I make 5 shots.
That's 6 choose 5 times 0.7.
So I'm only missing 1 shot, right?
So the probability of missing 1 is 0.7 and then I have to make
5, not necessarily in a row-- 0.3 to the fifth.
But you see, any given way that I make exactly 5 shots,
this is the probability.
And there are this many ways, 6 choose 5 ways of
making exactly 5 shots.
I could get the first 5 shots, miss the sixth.
I could get the last 5 shots.
I could make the first, miss the second, et
cetera and so forth.
And then finally, the probability that
X is equal to 6?
I make all of the shots.
That's 6 choose 6.
How many ways can I pick 6 things out of 6 choices?
And there's really only one way.
You could calculate it by calculating that.
Once again, you'll have to know that 0 factorial is equal to 1.
I have to make all 6 shots.
So 0.3 to the sixth.
0.3 times 0.3 times 0.3 times 0.3-- anyway, I'm
all out of time again.
In the next video we're going to graph this.