Binomial distribution
Binomial Distribution 3 Basketball binomial distribution
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- 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 MAKE (Sal says accidentaly wrong)
- 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.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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