Poker Hands Probability Explained Average ratng: 4,4/5 74 reviews
  1. Probability is the branch of mathematics that deals with the likelihood that one outcome or another will occur. For instance, a coin flip has two possible outcomes: heads or tails. The probability that a flipped coin will land heads is 50% (one outcome out of the two); the same goes for tails.
  2. The number of 5-card poker hands comprised of a full house can be computed using the multiplication principle. A full house can be considered a two-stage hand, the first stage being the pair, the second stage being the three-of-a-kind. Thus the number of possible full house hands = (the number of pairs) x (the number of threes-of-a-kind).
  3. Poker problems are very common in probability, and are harder than the simple question types mentioned above. The most common type of poker question involves choosing a five cards from the pack and asking the student to find the probability of a certain arrangement, called a poker hand. The most common arrangements are discussed in this section.

Sanderson M. Smith

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Poker odds with wild cards. Last week I wrote about the odds and probabilities of every five card poker hand. If you missed the article you can read it here. This week, I’m going to look at how these odds change if we add a wild card into the mix. A wild card, sometimes called a joker, can be used to represent a card of any value and suit.

POKER PROBABILITIES (FIVECARD HANDS)

In many forms of poker, one is dealt 5 cards from astandard deck of 52 cards. The number of different 5 -card pokerhands is

52C5 = 2,598,960

A wonderful exercise involves having students verify probabilitiesthat appear in books relating to gambling. For instance, inProbabilities in Everyday Life, by John D. McGervey, one findsmany interesting tables containing probabilities for poker and othergames of chance.

This article and the tables below assume the reader is familiarwith the names for various poker hands. In the NUMBER OF WAYS columnof TABLE 2 are the numbers as they appear on page 132 in McGervey'sbook. I have done computations to verify McGervey's figures. Thiscould be an excellent exercise for students who are studyingprobability.

There are 13 denominations (A,K,Q,J,10,9,8,7,6,5,4,3,2) in thedeck. One can think of J as 11, Q as 12, and K as 13. Since an acecan be 'high' or 'low', it can be thought of as 14 or 1. With this inmind, there are 10 five-card sequences of consecutive dominations.These are displayed in TABLE 1.

TABLE 1
A K Q J 10
K Q J 10 9
Q J 10 9 8
J 10 9 8 7
10 9 8 7 6
9 8 7 65
8 7 6 54
7 6 5 4 3
6 5 4 3 2
5 4 3 2 A
Poker Hands Probability Explained

The following table displays computations to verify McGervey'snumbers. There are, of course , many other possible poker handcombinations. Those in the table are specifically listed inMcGervey's book. The computations I have indicated in the table doyield values that are in agreement with those that appear in thebook.

TABLE 2
CalculatorProbability of poker hands explained
HAND

N = NUMBER OF WAYS listed by McGervey

Computations and comments
Probability of HAND
N/(2,598,960)
and approximate odds.

Straight flush

40

There are four suits (spades, hearts, diamond, clubs). Using TABLE 1,4(10) = 40.

0.000015
1 in 64,974

Four of a kind

624

(13C1)(48C1) = 624.

Choose 1 of 13 denominations to get four cards and combine with 1 card from the remaining 48.

0.00024
1 in 4,165

Full house

3,744

(13C1)(4C3)(12C1)(4C2) = 3,744.

Choose 1 denominaiton, pick 3 of 4 from it, choose a second denomination, pick 2 of 4 from it.

0.00144
1 in 694

Flush

5,108

(4C1)(13C5) = 5,148.

Choose 1 suit, then choose 5 of the 13 cards in the suit. This figure includes all flushes. McGervey's figure does not include straight flushes (listed above). Note that 5,148 - 40 = 5,108.

0.001965
1 in 509

Straight

10,200

(4C1)5(10) = 45(10) = 10,240

Using TABLE 1, there are 10 possible sequences. Each denomination card can be 1 of 4 in the denomination. This figure includes all straights. McGervey's figure does not include straight flushes (listed above). Note that 10,240 - 40 = 10,200.

0.00392
1 in 255

Three of a kind

54,912

(13C1)(4C3)(48C2) = 58,656.

Choose 1 of 13 denominations, pick 3 of the four cards from it, then combine with 2 of the remaining 48 cards. This figure includes all full houses. McGervey's figure does not include full houses (listed above). Note that 54,912 - 3,744 = 54,912.

0.0211
1 in 47

Exactly one pair, with the pair being aces.

84,480

(4C2)(48C1)(44C1)(40C1)/3! = 84,480.

Choose 2 of the four aces, pick 1 card from remaining 48 (and remove from consider other cards in that denomination), choose 1 card from remaining 44 (and remove other cards from that denomination), then chose 1 card from the remaining 40. The division by 3! = 6 is necessary to remove duplication in the choice of the last 3 cards. For instance, the process would allow for KQJ, but also KJQ, QKJ, QJK, JQK, and JKQ. These are the same sets of three cards, just chosen in a different order.

0.0325
1 in 31

Two pairs, with the pairs being 3's and 2's.

1,584

McGervey's figure excludes a full house with 3's and 2's.

(4C2)(4C1)(44C1) = 1,584.

Choose 2 of the 4 threes, 2 of the 4 twos, and one card from the 44 cards that are not 2's or 3's.

0.000609
1 in 1,641

Probability Of Poker Hands Explained

'I must complain the cards are ill shuffled 'til Ihave a good hand.'

-Swift, Thoughts on Various Subjects

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