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## Probability using combinatorics

# Probability & combinations (2 of 2)

## Video transcript

Welcome back. I actually recorded this video
earlier today, but then I realized my microphone
wasn't plugged in. And I won't name names in
terms of who unplugged it. But anyway, back
to probability. My wife is giggling
mischievously. Anyway, so let's do a
slightly harder problem than we did before. We were dealing with fair
coins, let's deal with a slightly unfair coin. Let's say I have a coin and
it's-- actually instead of unfair coin let's
do basketball. Let's say I'm shooting free
throws and I have a free throw percentage of 80%. So when I shot a free throw,
8 out of 10 times, or 80% of the time I will make it. But that also says that 20%
of time I will miss it. So given that, if I were to
take-- I don't know-- 5 free throws, what is the probability
that I make at least 3 of the 5 free throws? Well, let's think of it this
way, what is the probability of any particular combination
of making 3 out of the 5? So what do I mean by that? Let me pick a particular
combination. Let's say it's a basket,
basket, basket, and then I miss, miss. So that would be I
made 3 out of the 5. It could be-- I don't know--
basket, miss, basket, miss, basket. And there's a bunch of them and
we'll actually try to figure out how many of them there are. But what is the probability of
this particular combination? Well, I have an 80% chance of
making this first basket. Times 80%. That's a times right there. Times 80%. And then what's my
probability of missing? Well, that's 20%, right? Times 0.2. times 0.2. So this sequels 0.8 to the
third power times 0.2 squared. What's the probability of
getting this exact combination? Well, it's 0.8
times-- then I miss. There's a 20% chance of that. So times 0.2 times 0.8
times 0.2 times 0.8. We could rearrange this because
when you multiply numbers it doesn't matter what order
you multiply them in. So this is the same thing
as 0.8 times 0.8 times 0.8 times 0.2 times 0.2. So this is also the same
thing as 0.8 to the third times 0.2 squared. The probability of getting any
particular combination of 3 baskets and 2 misses is going
to be 0.8 to third times 0.2 squared. Now what's the total
probability of getting 3 out of 5? Well, it's going to be the sum
of all of these combinations. You know, I could list them
all, but we hopefully now are proficient enough in
combinatorics and combinations to figure out how many
different ways, if we have 5 baskets and we're picking-- or
we have 5 shots and we're picking 3 of them to be the
ones that are basket shots. So what do I mean? So let's say my 5 shots-- you
know, I've shot 1, 2, 3, 4, 5. Out of these five, I'm
going to choose 3. So once again, I'm putting my
hat on as the god of probability and I will choose 3
of these shots to be the ones that happen to be the
ones that get made. So essentially, out of
5 I am choosing 3. 5 choose 3. And what does that equal to? That's 5 factorial over 3
factorial times 5 minus 3 factorial, so that's
2 factorial. And that equals 5 times 4 times
3 times 2 times 1 over 3 times 2 times 1 times 2 times 1. We can ignore all the 1's. Let's see. We get 3 times 2 times 1. 3 times 2 times 1. We can cancel that. This 1 we can ignore, and
then this 2 and then this turns into 2. So there are 10
possible combinations. These are two of them. You know, basket, basket,
basket, miss, miss. Basket, miss, basket,
miss, basket. And you know, it's a good
exercise for you to list the other 8 of them. But using just the binomial
coefficient, and hopefully you have an intuition of why that
works and I'd be happy to make more videos if you feel that
that you need more explanation. But I made a couple. There are 10 combinations. So essentially, the probability
of getting exactly 3 out of 5 baskets, if I am an 80% free
throw shot, is going to be-- switch colors. The probability of 3 out of 5
baskets is going to be equal to the probability of each of the
combinations, which is 0.8 to the third times 0.2 squared. I make 3, miss 2. And then, times the total
number of combinations. Each of these has a
probability of this much. And then there's 10 different
arrangements that I could make. There's 10 different ways of
getting 3 baskets and 2 misses. So times 10, and what
does that equal to? Let me get my high-end
calculator here. So let's see what that is. That is 0.8 times 0.8 times 0.8
times 0.2 times 0.2 times 10. Equals 20.48. So it's essentially, a 20.48%
chance that I get exactly 3 out of 5 of the baskets. Now let's make it slightly
more interesting. Let's say I don't want it as a
probability of 3 out of the 5. And this is actually something
that probably, people are more likely to ask. What is the probability of
getting at least 3 baskets? Well, if you think about it,
this is the probability. This is equal to the
probability of getting 3 out of 5 baskets, plus the probability
of getting exactly 4 out of 5 baskets, plus the probability
of getting exactly 5 out of 5 baskets. We already figured
this one out. That's 20.48%. So what's the probability of
getting 4 out of 5 baskets? Well, once again, if we want
exactly 4 out of 5 baskets, so an example could be-- I
don't know-- miss, basket, basket, basket, basket. What's the probability of
any one of the combinations where I make 4 baskets? Well, it's going to be 0.8 to
the fourth times-- and then I have a 20% chance
that one miss. You know, it could have
been basket, miss, basket, basket, basket. That's also exactly 4, but
when you multiply them, the probability of getting any
one of these particular combinations is exactly this--
0.8 to the fourth times 0.2. If I have 5 baskets, how many
ways can I pick 4 of them to be the ones that I make if I'm
once again the god of probability? So this is going to be 0.8 to
the fourth times 0.2 times 0.2 times-- out of 5 baskets, I'm
choosing 4 that I'm going to make. So this is the number of
combinations where I get 4 out of the 5. So what is 5 choose 4? That's 5 factorial over 4
factorial times 1 factorial. Well, that equals just 5. You can work that out. So let's just figure this out. This is going to be 0.8
times 0.8 times 0.8-- that's 3-- times 0.8. That equals-- did
I do that right? Let's see. 0.1. Wait. 0.8 times 0.8--
yeah, that's right. Times 0.2 times 5. So 40.96%. So this is 40.96%. So roughly, 41% chance that I
get exactly 4 out of 5 baskets. Which is interesting because
that's kind of my free throw percentage. So it's almost a little less--
you know, 2/3 shot of kind of hitting my free throw
percentage on the mark on that time. And what's the probability
of getting 5 out of 5 Well, there's only one way
of getting 5 out of 5. You have to get all 5 of them. So this is 0.8 to
the fifth power. Let me get the calculator back. So it's 0.8 times 0.-- oh,
wait-- times 0.8 times 0.8 times 0.8 times 0.8
equals 0.3276. So 32.77% shot. And then we can add them
all up because we want the probability of at least 3. So it's going to be that, the
probability of getting 5 out of 5, plus the probability of
getting 4 out of 5, which is 0.4096. Plus, the probability
of getting 3 out of 5. So that's 0.2048
equals 0.94208. So 94.21-- roughly, rounding--
% chance, which makes sense. If I have an 80% free throw
percentage on any one shot, I have a very high probability of
getting at least 3 out of 5 when I go to the
free throw line. Anyway, I'm all out of time. I'll see you in the next video.