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Poker Winning Hand Odds in 5 card Games -

There are many poker variants, but unless otherwise specified in the rules of the variant being played, hands are evaluated using the traditional set of five-card hands. These are, from worst to best:

No pair (for example, A-Q-10-5-2)

One pair (for example, 10-10-K-7-4)

Two pair (for example, K-K-9-9-J)

Three of a kind (for example, 4-4-4-K-9)

Straight (for example, 9-8-7-6-5)

Flush (for example, K-J-8-7-4 of one suit)

Full house (for example, 2-2-2-Q-Q)

Four of a kind (for example, J-J-J-J-6)

Straight flush (for example, Q-J-10-9-8 of one suit)

Royal straight flush (only A-K-Q-J-10 of a single suit; not technically a separate hand, but sans wild cards the highest possible in poker [also called simply a royal flush])

Five of a kind (for example, 6-6-6-6-(Wild), only possible in Wild card game. Poker games may contain one or more cards designated as wild . These may be jokers, or they may be normal ranked and suited cards pressed into wild card duty ("deuces wild" is a common variant).

These hands are called the traditional hands or high hands.

The odds chart below says it all, pretty much.
The probability distribution of the various hands for a 5-card game without wild cards and playing Ace high only is as follows:

Hand

Probability

Distribution

No pair

1,303,560 in 2,598,960

50

.16%

One pair

1,098,240 in 2,598,960

42

.26%

Two pair

123,552 in 2,598,960

4

.75%

Three of a kind

54,912 in 2,598,960

2

.11%

Straight

9,180 in 2,598,960

0

.353%

Flush

5,112 in 2,598,960

0

.197%

Full house

3,744 in 2,598,960

0

.144%

Four of a kind

624 in 2,598,960

0

.0240%

Straight flush (excluding Royal flush)

32 in 2,598,960

0

.00123%

Royal straight flush

4 in 2,598,960

0

.000154%

TOTAL

2,598,960 in 2,598,960

100

.00%

If Ace is played low as well as high, then the probabilities of the following hands change:

Hand

Probability

Distribution

No pair

1,302,540 in 2,598,960

50

.12%

Straight

10,200 in 2,598,960

0

.392%

Flush

5,108 in 2,598,960

0

.197%

Straight flush (excluding Royal flush)

36 in 2,598,960

0

.00139%

Why 2,598,960? That is the number of total hands that can be dealt in 5 card poker, excluding permutations of the same hand.

Here's another look at the odds, with 5 and 7 card hands compared. These numbers are from actual hand histories, which will make these odds very slightly different from the theoretical calculated ones. Notice how much easier it is to make a given hand when you have 7 cards to work with.

Note: Standard rankings are used here, lowest to highest, with no wilds.

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