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 poker odds calculator using efficient, correct and
optimized algorithm.
Odds
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).
Odds 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