Combinatorics and probability
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To win a particular lottery game, a player chooses 4 numbers from 1 to 60. Each number can only be chosen once. If all 4 numbers match the 4 winning numbers, regardless of order, the player wins. What is the probability that the winning numbers are 3, 15, 46, and 49? So the way to think about this problem, they say that we're going to choose four numbers from 60. So one way to think about it is, how many different outcomes are there if we choose four numbers out of 60? Now this is equivalent to saying, how many combinations are there if we have 60 items? In this case we have 60 numbers, and we are going to choose four. And we don't care about the order. That's why we're dealing with combinations, not permutations. We don't care about the order. So how many different groups of four can we pick out of 60? And we don't care what order we picked them in. And we've seen in previous videos that there is a formula here, but it's important to understand the reasoning behind the formula. I'll write the formula here, but we'll think about what it's actually saying. So this is 60 factorial over 60 minus 4 factorial, divided also by 4 factorial, or the denominator multiplied by 4 factorial. So this is the formula right here. But what this is really saying, this part right here, 60 factorial divided by 60 minus 4 factorial, that's just 60 times 59, times 58, times 57. That's what this expression right here is. And if you think about it, the first number you pick-- there's 1 of 60 numbers, but then that number is kind of out of the game. Then you can pick from 1 of 59, then from 1 of 58, then of 1 of 57. So if you cared about order, this is the number of permutations. You could pick four items out of 60 without replacing them. Now, this is when you cared about order, but you're overcounting because it's counting different permutations that are essentially the same combination, essentially the same set of four numbers. And that's why we're dividing by 4 factorial here. Because 4 factorial is essentially the number of ways that four numbers can be arranged in four places. Right? The first number can be in one of four slots, the second in one of three, then two, then one. That's why you're dividing by 4 factorial. But anyway, let's just evaluate this. This'll tell us how many possible outcomes are there for the lottery game. So this is equal to-- we already said the blue part is equivalent to 60 times 59, times 58, times 57. So that's literally 60 factorial divided by essentially 56 factorial. And then you have your 4 factorial over here, which is 4 times 3, times 2, times 1. And we could simplify it a little bit just before we break out the calculator. 60 divided by 4 is 15. And then let's see, 15 divided by 3 is 5. And let's see, we have a 58 divided by 2 is 29. So our answer is going to be 5 times 59, times 29, times 57. Now this isn't going to be our answer. This is going to be the number of combinations we can get if we choose four numbers out of 60 and we don't care about order. So let's take the calculator out now. So we have 5 times 59, times 29, times 57. It's equal to 487,635. So let me write that down. That is 487,635 combinations. If you're picking four numbers, you're choosing four numbers out of 60, or 60 choose four. Now, the question they say is, what is the probability that the winning numbers are 3, 15, 46, and 49? Well, this is just one particular of the combinations. This is just one of the 487,635 possible outcomes. So the probability of 3, 15, 46, 49 winning is just equal to-- well, this is just one of the outcomes out of 487,635. So that right there is your probability of winning. This is one outcome out of all the potential outcomes or combinations when you take 60 and you choose four from that.