Probability and Poker

cards

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: ♥ 2 3 4 5 6 7 8 9 10 J Q K A
13 diamonds: ♦ 2 3 4 5 6 7 8 9 10 J Q K A
13 clubs: ♣ 2 3 4 5 6 7 8 9 10 J Q K A
13 spades: ♠ 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 .

Number of possible poker hands = C525 = 52!/(5!×47!) = 2,598,960.

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

4/2,598,960, or 0.000002.

Straight Flush

The next most valuable type of hand is a 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 ones starting with Ace, 1, 2, ... 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

40/2,598,960, or 0.000015.

The table below lists the number of possible ways that different types of hands can arise and their probability of occurrence.

Ranking, Frequency and Probability of Poker Hands

Hand No. of Ways Probability Description
Royal Flush

4

0.000002

Ten, J, Q, K, A of one suit.
Straight Flush

40

0.000015

A straight is 5 cards in order.
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.
For example, 2, 4, 5, 9, J (all hearts) is a flush.
Straight

10,200

0.003925

The 5 cards are in order.
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
Nothing

1,302,540

0.501177

Example: 3, 6, 8, 9, K (at least two different suits)

Question

The probability for a full house is given above as 0.001441. Where does this come from?

Answer




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