Poker Probability

Frequency of 5 card poker hands

Derivation

• Straight flush -- Each straight flush is uniquely determined by its highest ranking card; and these ranks go from 5 (A-2-3-4-5) up to A (T-J-Q-K-A) in each of the 4 suits. Thus, the total number of straight flushes is

$\{40\; \backslash choose\; 1\}\; =\; 40$

• Four of a kind -- First, we choose one of the 13 ranks for the 4 of a kind; then there are 52 − 4 = 48 cards remaining from which to choose the final card. Thus, the total number of four of a kinds is

$\{13\; \backslash choose\; 1\}\{48\; \backslash choose\; 1\}\; =\; 624$

• Full house -- First, we choose one of the 13 ranks and 3 of the 4 suits for the 3 of a kind; then we choose one of the remaining 12 ranks and 2 of the 4 suits for the pair. Thus, the total number of full houses is

$\{13\; \backslash choose\; 1\}\{4\; \backslash choose\; 3\}\{12\; \backslash choose\; 1\}\{4\; \backslash choose\; 2\}\; =\; 3,744$

• Flush -- First, we choose one of four suits; then we choose 5 of the 13 possible ranks. Finally, we must subtract the 40 straight flushes, since these are ranked as straight flushes, not flushes. Thus, the total number of flushes is

$\{4\; \backslash choose\; 1\}\{13\; \backslash choose\; 5\}\; -\; 40\; =\; 5,108$

• Straight -- First, we choose the highest ranking card; there are 10 of these, from 5 (A-2-3-4-5) to A (T-J-Q-K-A). Then we choose one of four suits for each of the 5 cards. Finally, we must subtract the 40 straight flushes, since these are ranked as straight flushes, not straights. Thus, the total number of straights is

$\{10\; \backslash choose\; 1\}\{4\; \backslash choose\; 1\}^5\; -\; 40\; =\; 10,200$

• Three of a kind -- First, we choose one rank out of 13 for the 3 of a kind; then we choose 3 out of 4 suits for the 3 of a kind. Then we choose 2 distinct ranks out of the remaining 12 for the other 2 cards, as well as suits for each of those cards. Thus, the total number of three of a kinds is

$\{13\; \backslash choose\; 1\}\{4\; \backslash choose\; 3\}\{12\; \backslash choose\; 2\}\{4\; \backslash choose\; 1\}^2\; =\; 54,912$

• Two pair -- First, we choose 2 of the 13 ranks for the 2 pairs; then we choose 2 out of 4 suits for each of those 2 pairs. The final card can be any one of the 44 remaining cards not comprising the ranks of the 2 pairs. Thus, the total number of two pairs is

$\{13\; \backslash choose\; 2\}\{4\; \backslash choose\; 2\}^2\{44\; \backslash choose\; 1\}\; =\; 123,552$

• One pair -- First, we choose one of the 13 ranks for the pair; then we choose 2 out of 4 suits for that pair. For the other 3 cards, we choose 3 ranks out of the remaining 12 and one of 4 suits for each of the 3 cards. Thus, the total number of one pairs is

$\{13\; \backslash choose\; 1\}\{4\; \backslash choose\; 2\}\{12\; \backslash choose\; 3\}\{4\; \backslash choose\; 1\}^3\; =\; 1,098,240$

• No pair -- We can choose 5 out of 13 ranks, discounting the 10 possible straights. Then we choose one of 4 suits for each of the 5 cards, discounting the 4 possible flushes. Alternatively, since we are looking for any hand which does not fall into one of the above categories, we can take the total number of 5 card hands and subtract the sum of the above. Thus, the total number of no pair hands is

$\backslash left\backslash choose\; 5\}\; -\; 10\backslash right(4^5\; -\; 4)\; =\; \{52\; \backslash choose\; 5\}\; -\; 1,296,420\; =\; 1,302,540$

External links

• MathWorld: Poker
• Probability and Poker
• Brian Alspach's mathematics and poker page
• Brian Alspach Poker Math articles
• Brian Alspach Poker Math computations: 3-Card Poker Hands, 4-Card Poker Hands, 5-Card Poker Hands, 6-Card Poker Hands, 7-Card Poker Hands, 7-Card Poker Hands with Joker, Sousem Hands, Mambo Poker, Low Board Probability, Flushing Boards: I, Flushing Boards: II, Flushing Boards and Random Hands, Computations for Losing Flushes, Overcards in Hold'Em, Straight Completion, 7-Card Stud Flush Completion, Suit and Rank Distributions for Boards, Bad Beat Jackpots: Omaha, Multiple Pocket Pairs, Rank Sets and Straights, Pai Gow Poker Computations, Flop Sums, Enumerating Starting Poker Hands, Probabilities for Successive Memorable Hands, Online Bad-Beat Jackpot Computations

<< Previous Word Browser Next >>
abstract syntax abstract syntax notation one jean chrtien abstract syntax tree ast nidhogg las vegas (disambiguation) petra kelly minimalist trance carl hiaasen sauna	lituus hyperthermia game classification battle of almansa sweat lodge list of chess world championship matches communications in saint kitts and nevis betting (poker) list of poker variants five card stud seven card stud	testimony texas hold 'em sick puppy list of miscellaneous poker variants ega sandbagging (poker) poker psychology cheating in poker conciergerie public cardroom rules (poker) low hand (poker)	high hand jean pierre jeunet list of colleges and universities list of colleges and universities by country list of colleges and universities starting with a list of colleges and universities starting with b list of colleges and universities starting with c list of colleges and universities starting with d list of colleges and universities starting with e list of colleges and universities starting with f list of colleges and universities starting with g