Poker is also a game of mathematics and all odds and winning
chances can be calculated in a fraction of a second.
Playing poker without knowing probabilities decreases massively player's chance of winning.
This site provides theoretical explanations of poker probability and the probability calculator using efficient, correct and optimized algorithm.
If an event has n possible outcomes and a particular result has m outcomes, then the probability of the particular result is m/n. The probability of drawing a specific 5-card hand means the division of the number of ways of drawing such a hand by the number of all hands.
Poker is played with a 52 card deck and the total number of 5-card hands is 2,598,960 (the number of combinations of 5 cards chosen from 52 cards).
Probabilities of 5-card poker hands
Royal Flush - There are only 4 royal flush hands, so that the probability of a royal flush is 4/2,598,960.
Straight Flush - There are 10 different straight flushes of each suit, but the highest type of straight flush is a royal flush, so the probability of getting a straight flush is = (10-1)*4 / 2,598,960 = 36/2,598,960
Four of a Kind - There are 13 possible distinct sets of four of a kind and the 5th card can be any of the other 48 cards, so the probability of four of a kind is = 13*48 / 2,598,960 = 624/2,598,960
Full House - The full house consists of a 3 of a kind (triple) and a pair. There are 4 suits and 13 ranks. Each triple has 3 different suits. The number of all triples of each rank is comb (4;3) = 4. The number of all triples = 4*13 = 52. The number of all pairs of each rank is comb(4;2) = 6. The number of all pairs = 6*12 = 72 (12 = the number all ranks without the rank of the current triple). In the end we get the probability of full house as 52*72 / 2,598,960 = 3,744/2,598,960
Flush - There are 5,148 (comb(13;5)*4) of poker hands in one suit only. We do not want to count the royal flush and the straight flush to this group, so we calculate the probability of getting flush as: 5,148-36-4 / 2,598,960 = 5,108/2,598,960
Straight - There are 10 different straight types, each type starting with the different card rank. Every card of a straight hand can have one of four suits. So the number of all straight hands is = 10*4^5 = 10,240. As royal flush and straight flush are also straights, we get the probability of straights as 10,240 - 4 - 36 / 2,598,960 = 10,200/2,598,960
Three of a Kind - Each triple has 3 different suits out of 4 possible suits. The number of all triples of each rank is comb(4;3) = 4. The number of all triples = 4*13 = 52. The remaining two cards can have any two of the remaining twelve ranks (without having the same rank) and any of the 4 suits. The probability of getting three of a kind is 13*comb (4;3)* comb(12;2)*4*4 = 54,912/2,598,960
Two Pairs - The pairs ranks can be any combination of 2 ranks chosen from all 13 ranks, both pairs can have 2 from all 4 suits. The 5th. card must have a rank not used by pairs and can have any of the 4 suits. The probability of getting two pairs is comb(13;2)*comb(4;2)^2*11*4 = 123,552/2,598,960
One Pair - The pair rank can be any of possible 13 ranks and can have 2 from all of 4 suits. The remaining 3 cards can have any combination of 3 ranks chosen from remaining 12 ranks. Each card of the remaining 3 cards can have any of the 4 suits. The probability of getting one pair is 13*comb(4;2)*comb(12;3)4*4*4 = 1,098,240/2,598,960
High Card - The probability of getting nothing is 1 minus the probability of getting something = 1,302,540/2,598,960