Example: A Poker Hand

The game of poker has many variants. Common to all is the fact that players get - one way or another - hands of five cards each. The hands are compared according to a predetermined ranking system. Below, we shall evaluate probabilities of several hand combinations.

Poker uses the standard deck of 52 cards. There are C(52, 5) possible combinations of 5 cards selected from a deck of 52: 52 cards to choose the first of the five from, 51 cards to choose the second one, ..., 48 to choose the fifth card. The product 52×51×50×49×48 must be divided by 5! because the order in which the five cards are added to the hand is of no importance, e.g., 7♣8♣9♣10♣J♣ is the same hand as 9♣7♣10♣J♣8♣. Thus there are C(52, 5) = 2598960 different hands. The poker sample space consists of 2598960 equally probable elementary events.

The probability of whichever hand is naturally 1/2598960. [Mazur, pp. 81-82] shows another elegant way of arriving at the same probability. Imagine having a urn with 52 balls, of which 5 are black and the remaining white. You are to draw 5 balls out of the urn. What is the probability that all 5 balls drawn are black?

The probability that the first ball is black is 5/52. Assuming that the first ball was black, the probability that the second is also black is 4/51. Assuming that the first two balls are black, the probability that the third is black is 3/50, ... The fifth ball is black with the probability of 1/48, provided the first 4 balls were all black. The probability of drawing 5 black balls is the product:

C(52,5)

The highest ranking poker hand is a Royal Flush - a sequence of cards of the same suit starting with 10, e.g., 10♣J♣Q♣K♣A♣. There are 4 of them, one for each of the four suits. Thus the probability of getting a royal flush is 4/2598960 = 1/649740. The probability of getting a royal flush of, say, spades ♠, is of course 1/2598960.

Any sequence of 5 cards of the same suit is a straight flush ranked by the highest card in the sequence. A straight flush may start with any of 2, 3, 4, 5, 6, 7, 8, 9, 10 cards and some times with an Ace where it is thought to have the rank of 1. So there are 9 (or 10) possibilities of getting a straight flush of a given suit and 36 (or 40) possibilities of getting any straight flush.

Five cards of the same suit - not necessarily in sequence - is a flush. There are 13 cards in a suit and C(13, 5) = 1287 combinations of 5 cards out of 13. All in all, there are 4 times as many flush combinations: 5148.

Four of a kind is a hand, like 5♣5♠5♦5♥K♠, with four cards of the same rank and one extra, unmatched card. There are 13 combinations of 4 equally ranked cards each of which can complete a hand with any of the remaining 48 cards. Giving the total of 13×48 = 624 possible "four of a kind" combinations.

A hand with 3 cards of one rank and 2 cards of a different rank is known as Full House. For a given rank, there are C(4, 3) = 4 ways to choose 3 cards of that rank; there 13 ranks to consider. There are C(4, 2) = 6 combinations of 2 cards of equal rank, but now only 12 ranks to choose from. There are then 4×13×6×12 = 3744 full houses.

A straight hand is a straight flush without "flush", so to speak. The card must be in sequence but not necessarily of the same suit. If the ace is allowed to start a hand, there are 40 ways to choose the first card and then, we need to account that the remaining 4 cards could be of any of the 4 suits, giving the total of 40×4×4×4×4 = 10240 hands. Discarding 40 straight flushes leaves 10200 "regular" flushes.

Three of a kind is a hand, like 5♣5♠5♦7♥K♠, where three cards have the same rank while the remaining 2 differ in rank between themselves and the first three. There are 13×C(4, 3) = 52 combinations of three cards of the same rank. The next card could be any of 48 and the fifth any of 44 and the pair could come in any order so the products needs to be halved: 52×48×44 / 2 = 54912.

There remain Two pair and One pair combinations that are left as an exercise.

References

J. Mazur, What's Luck Got to Do with It?, Princeton University Press, 2010

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