Chapter Contents

• Counting and Probability - Introduction
• 1. Factorial Notation
• 2. Basic Principles of Counting
• 3. Permutations
• 4. Combinations
• 5. Introduction to Probability Theory
• 6. Probability of an Event
• Singapore TOTO

• Probability and Poker

• 7. Conditional Probability
• 8. Independent and Dependent Events
• 9. Mutually Exclusive Events
• 10. Bayes’ Theorem
• 11. Probability Distributions - Concepts
• 12. Binomial Probability Distributions
• 13. Poisson Probability Distribution
• 14. Normal Probability Distribution
• The z-Table
• Counting and Probability Problem Solver

• Comments, Questions?

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Probability and Poker

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 `C_r^n`.

The number of possible poker hands

`=C_5^52=(52!)/(5!xx47!)=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)=0.000\ 002`

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 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/(2,598,960)=0.000\ 015\ 39`

[Note: There is some overlap here since the straight flush starting at 10 is the same as the royal flush. So strictly there are 36 straight flushes (4 × 9) if we don't count the royal flush. The probability of getting a straight flush then is 36/2,598,960 = 0.00001385.]

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	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) is a 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
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