Probability & combinations (2 of 2)

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.