Probabilities in Poker

Probabilities of different poker hands

Below are the probabilities of being dealt the following poker hands. In mathematics a square is a special case of a rectangle, however, in poker a different convention applies: a flush does not include straight flushes, and straight flushes do not include royal flushes. For example, a flush is when all 5 cards belong to the same suit and the cards are not consecutive. Thus the hand 2H,3H,4H,5H,6H is a straight flush and not a flush. Similarly, 10H,JH,QH,KH,AH is a royal flush and not a straight flush. This complicates the counting somewhat, however, it is more useful for gambling: as royal flush beats a straight flush which in turn beats a flush. In general, less probable poker hands beat more probable hands.

 
                                           Approximate            
Poker Hand       Probability of Hand       Probability (2 d.p. denominators)
---------------  ----------------------   ------------
Royal Flush:             4 in 2,598,960 =  1/649,740 
Straight Flush:         36 in 2,598,960 =  1/72,193.33
Four of a Kind:        624 in 2,598,960 =  1/4,165
Full House:          3,744 in 2,598,960 =  1/694.17
Flush:               5,108 in 2,598,960 =  1/508.80
Straight:           10,200 in 2,598,960 =  1/255.8
Three of a Kind:    54,912 in 2,598,960 =  1/47.33
Two Pair:          123,552 in 2,598,960 =  1/21.04
One Pair:        1,098,240 in 2,598,960 =  1/2.37
Nothing:         1,302,540 in 2,598,960 =  1/1.96


Notation and Terminology

Each card has 13 different ranks (or face values) namely 2,3,4,5,6,7,8,9,10,J,Q,K,A where J,Q,K,A stand for Jack, Queen, King and Ace. The 4 different suits are denoted by H,D,C,S. Thus AH and 2S denote the Ace of Hearts, and the 2 of Spades respectively. The 13 ranks times 4 suits gives 52 cards. The number of k-element subsets of a set such as {1,2,3,...,n} with n elements is (n choose k) = n!/(k!*(n-k)!). As usual n!, pronounced "n factorial", is the product n(n-1)(n-2)...2.1. For example five factorial, denoted 5!, is 5*4*3*2*1 = 120.

The number of ways to choose 5 cards from a deck of 52 cards is (52 choose 5) = 2,598,960. The probability of being dealt any type of poker hand is the number of poker hands of this type divided by the total number of poker hands (namely 2,598,960). We count below the number of poker hands of a given type.

Royal Flush

There are 4 different royal flushes: one for each suit. Precisely they are: 10H,JH,QH,KH,AH; 10D,JD,QD,KD,AD; 10C,JC,QC,KC,AC; 10S,JS,QS,KS,AS.

Straight Flush

The highest card in a straight flush can be 5,6,7,8,9,10,J,Q, or K. (Remember an Ace can represent a 1 or a 13 and so can be at the beginning or the end of a flush, however, as we do not want to count royal flushes so we do not allow Ace to represent 13.) Thus there are 9 possible high cards, and 4 possible suits and thus 9 * 4 = 36 different possible straight flushes.

Four of a Kind

There are 13 different possible ranks of the 4 of a kind. The fifth card could be anything of the remaining 52 - 4 = 48. Thus there are 13 * 48 = 624 different four of a kinds.

Full House

The 3 of a kind can be chosen in 13 * (4 choose 3) ways as there are 13 possible ranks and ( 4 choose 3) ways to choose 3 cards from a given rank. The pair (or 2 of a kind) can be chosen in 12 * (4 choose 2) ways as there are 12 = 13 - 1 possible different ranks and (4 choose 2) ways to choose 2 cards from a given rank. In total, there are 13 * 4 * 12 * 6 = 3,744 ways to form a full house.

Flush

There are 4 suits and (13 choose 5) = 1,287 ways to choose five cards in the same suit. Thus there are 4 * 1,287 possibilities some of which are royal and straight flushes. Subtract the royal and straight flushes gives 4 * 1,287 - 4 - 36 = 5,108 flushes.

Straight

The highest card in a straight flush can be 5,6,7,8,9,10,J,Q,K, or A. Thus there are 10 possible high cards. Each card may be of four different suits. Subtract the royal and straight flushes gives 10 * 45 - 4 - 36 = 10,200 straights.

Three of a Kind

There are 13 ranks to choose from for the three of a kind and 4 ways to arrange 3 cards among the four to choose from. There are (12 choose 2) = 66 ways to arrange the other two ranks to choose from for the other two cards. In each of the two ranks there are four cards to choose from. Thus there are 13 * 4 * 66 * 42 = 54,912 ways to form a three of a kind. [Alternatively, the last two cards can be chosen in 48 * 44 / 2 ways. The fourth card must have a different rank to the three of a kind giving 52 - 4 = 48 possibilities, and the fifth card must be another different rank giving 52 - 4 = 48 possibilities. The order of the last two card is irrelevant so we must divide by 2.]

Two Pair

There are (13 choose 2) = 78 ways to arrange the two ranks represented. In both ranks there are (4 choose 2) = 6 ways to arrange two cards. There are 44 cards left for the fifth card. Thus there are 78 * 62 * 44 = 123,552 ways to form two pair.

One Pair

There are 13 ranks to choose from for the pair and (4 choose 2) = 6 ways to choose the two cards in the pair. There are (12 choose 3) = 220 ways to arrange the other three ranks of the singletons, and four cards to choose from in each rank. Thus there are 13 * 6 * 220 * 43 = 1,098,240 ways to arrange a pair.

Nothing

There must be five different ranks represented, of which there are (13 choose 5) = 1,287 possible combinations. Each rank has four cards to choose from. Finally subtract the number of straights, flushes, straight flushes, and royal flushes, to avoid double counting. Thus the number of ways to form nothing is 1,287 * 45 - 4 - 36 - 5,108 - 10,200 = 1,302,540.

Our 10 counting problems Royal Flush, Straight Flush, ..., Nothing exhaust all possibilities. If we add the number in each of the 10 categories we get 4+36+624+3744+5108+10200+54912+123552+1098240+1302540 which equals 2598960 or (52 choose 5). This strongly suggests that we have not made any errors.

Card Counting

Card counting is a strategy where you keep certain running totals in order to estimate your chances of winning. It does not involve memorizing cards, but it does require concentration especially in noisy environments. Card counting in a simpler game such as Blackjack is easier to understand than in Poker. Cassino owners know about card counting techniques, and generally invoke rules about shuffling or sitting out to increase the odds in their favor.