 So one common use of hypergeometric probabilities is in calculating the probabilities of poker hands. And this was great when everybody knew how to play the game of poker. But not everybody does, so here's a quick introduction, not to the game itself, but to the probability space we're talking about. In a standard deck of 52 cards, there are four suits, spades, clubs, hearts, and diamonds. Spades and clubs are black, hearts and diamonds are red. And each suit has 13 cards, which we can think about as being numbered 1, which is the ace, through 10, and then Jack, Queen, King. And given this setup of the cards, we can compute poker probabilities. For example, one of the most notorious hands in the game of poker is known as a dead man's hand, because according to well-documented myth, this was the hand that was drawn by Wildville Hickok when he was shot and killed in the middle of a poker game. And this consists of the black aces and the black aides, plus a fifth card. So if we assume that all sets of five cards are equally likely to occur, this is a combinatorial probability. So first we note there's 52 choose five, about 3 million possible draws of five cards. Now to select such a hand, what we need to do is we need to select the two black aces, select the two black aides, and select a fifth card. Now while we can make these selections in any order that we want to, it's helpful to work from the most specific to the least. So if I want to select those two black aces, well there's only two black aces in the deck. So I have to choose both of them. And so there's two, combined two, one way to select the two black aces. To select the two black aides, well there's again only two black aides so I have to select both of them. So there's two, combined two, one way to select the two black aides. Now, that 5th card, I've already selected a total of 4 cards, 2 black aces, 2 black 8s, and so there's only 48 cards left, and so there's 48 combined 1, 48 ways to select that 5th card, and these are permutations. I can't use the 2 black aces that I selected as the 5th card, and so I can multiply these three numbers together, 1 times 1, times 48, and so there's 48 possible dead man's hands, and so the probability of selecting that is about 1 in 54,000. How about some more conventional poker probabilities? So a flush consists of five cards in the same suit. What's the probability of drawing a flush? And so let's think about this. We choose a flush by choosing a suit, and there's four ways of choosing a suit. Now, if we want all of our cards to be from the same suit, we've got to choose five of the 13 cards in the suit, and so there's 13 combined 5, 1,287 ways to do that. And again, these choices form a permutation because we can't switch our answers around, so we can determine the number of flushes by multiplication. So again, there's 52 combined, about 3 million possible draws of five cards. There's 5,148 possible flushes, and so the probability of a flush is about 1 in 500. Now there's a number of poker variants, so instead poker, players receive seven cards and choose five. What's the probability of a flush, which is again still five cards of the same suit? Now, assuming we're playing to win, we would actually choose the five cards of the same suit that we got. And this means that our seven cards could have five, six, or seven cards of the same suit. So this time there's 52 combined seven, or about 133 million different ways to choose seven cards. Now to get five cards of the same suit and two cards of different suits, we select a flush, and there's still 5,148 possibilities. But we need to select two more cards of a different suit. And in particular, there's 52 minus 13, 39 cards of a different suit. And so we can pick two of them in 39 combined two, 741 ways. And again, because the flush that we select and the two cards of a different suit that we select are not interchangeable, we can multiply these two numbers and see that there are 51,48 times 741 hands of five cards of one suit and two of other suits. Now to get six cards of the same suit and one card of a different suit, we'll select a suit. There's four ways we can select a suit. We'll select six cards from the suit. That's 13 choose six. And that works out to be 1,716 ways. And so there's four times 17, 16, 6,864 ways to select six cards from one suit. And that last card can be any of the 39 cards left to choose from. And again, what we're describing is a permutation. And so there's 6,864 times 39 hands with six cards of one suit and one card of another suit. And finally, to get seven cards of the same suit, select a suit. Four ways to do that. Pick seven cards from the same suit. That's 13 combined seven. Again, 1,716 ways. And so we get four times 17, 16, 68, 64 ways. And so all together there are a lot of different ways of getting a flush. And so if all of our hands are equally likely, the probability of getting a flush is about 1 in 33.