List the poker hand types for standard 5-card poker, straight flush, four of a k

Question

List the poker hand types for standard 5-card poker, straight flush, four of a kind, full house, flush, straight, three of a kind, two pair, pair, and nothing. For each hand type, show how to calculate the number of hands, give the number of hands, and find the probability for that hand type.

Repeat #1 for 6-card poker (my invention) with hand types: 6-card straight flush, 5-card straight flush, four of a kind and two of a kind, two three of a kind, three pair, four of a kind, full house, 6-card flush, 5-card flush, 6-card straight, 5-card straight, three of a kind, two pair, pair, and nothing. List the hands in order of increasing number of possibilities (highest to lowest rank).

Explanation / Answer

In the standard game of poker, each player gets 5 cards and places a bet, hoping his cards are "better" than the other players' hands.

The game is played with a pack containing 52 cards in 4 suits, consisting of:

13 hearts:
13 diamonds
13 clubs:
13 spades:

2 3 4 5 6 7 8 9 10 J Q K A
2 3 4 5 6 7 8 9 10 J Q K A
2 3 4 5 6 7 8 9 10 J Q K A
2 3 4 5 6 7 8 9 10 J Q K A

The number of different possible poker hands is found by counting the number of ways that 5 cards can be selected from 52 cards, where the order is not important. It is a combination, so we use Cnr

The number of possible poker hands

=C525=52!/5!*47!=2598960

Straight flush;

which is 5 cards in order, all of the same suit.

For example, 2, 3, 4, 5, 6 is a straight flush.

For each suit there are 10 such straights (the one starting with Ace, the one starting with 2, the one starting with 3, through to the one starting at 10) and there are 4 suits, so there are 40 possible straight flushes.

The probability of being dealt a straight flush is

=40/2598960=0.00001385

Royal flush:

The best hand (because of the low probability that it will occur) is the royal flush, which consists of 10, J, Q, K, A of the same suit. There are only 4 ways of getting such a hand (because there are 4 suits), so the probability of being dealt a royal flush is

4/2598960=0.000001539

Ranking, Frequency and Probability of Poker Hands

0.000015

Example: 4 kings and any other card

Hand No.of ways Probability Description Royal flush 4 0.000002 Ten, J, Q, K, A of one suit. Straight Flush 36

0.000015

A straight is 5 cards in order.
(Excludes royal and straight flushes.)
An example of a straight flush is: 5, 6, 7, 8, 9, all spades. Four of a Kind 624 0.000240

Example: 4 kings and any other card

Full House
3,744
0.001441 3 cards of one denominator and 2 cards of another. For example, 3 aces and 2 kings is a full house. Flush 5,108 0.001965 All 5 cards are from the same suit.
(Excludes royal and straight flushes)
For example, 2, 4, 5, 9, J (all hearts) isa flush. Straight 10,200 0.003925 The 5 cards are in order.
(Excludes royal flush and straight flush)
For example, 3, 4, 5, 6, 7 (any suit) is a straight Three of a Kind 54,912 0.021129 Example: A hand with 3 aces, one J and one Q. Two Pairs 123,552 0.047539 Example: 3, 3, Q, Q, 5 One Pair 1,098,240 0.422569 Example: 10, 10, 4, 6, K

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