Games of chance: Poker

A deck of cards naturally has the structure of a product set and thus can be modeled mathematically by

D={1,2,3,4,5,6,7,8,9,10,j,q,k}×{♣,♢,♡,♠}

where the first coordinate represents the denomination or kind (ace, two through 10, jack, queen, king) and where the second coordinate represents the suit (clubs, diamond, hearts, spades). Sometimes we represent a card as a string rather than an ordered pair

There are many different poker games, but we will be interested in standard draw poker, which consists of dealing 5 cards at random from the deck D .

The order of the cards does not matter in draw poker, so we will record the outcome of our random experiment as the random set (hand)

X is X1,X2,X3,X4,X5
where Xi=(Yi,Zi)∈D 
for each i and Xi≠Xj 
for i≠j. 

Thus, the sample space consists of all possible poker hands:

S is x1,x2,x3,x4,x5 :xi∈D 
for each i and xi≠xj 
for all i≠j}

Our basic modeling assumption (and the meaning of the term at random) is that all poker hands are equally likely.

Thus, the random variable X is uniformly distributed over the set of possible poker hands S.

P(X∈A)=#(A)#(S)

In statistical terms, a poker hand is a random sample of size 5 drawn without replacement and without regard to order from the population D.

The Value of the Hand

There are nine different types of poker hands in terms of value. We will use the numbers 0 to 8 to denote the value of the hand, where 0 is the type of least value (actually no value) and 8 the type of most value.

The hand value V of a poker hand is a random variable taking values 0 through 8, and is defined as follows:

• No Value. The hand is of none of the other types.
• One Pair. The hand has 2 cards of one kind, and one card each of three other kinds.
• Two Pair. The hand has 2 cards of one kind, 2 cards of another kind, and one card of a third kind.
• Three of a Kind. The hand has 3 cards of one kind and one card of each of two other kinds.
• Straight. The kinds of cards in the hand form a consecutive sequence but the cards are not all in the same suit. An ace can be considered the smallest denomination or the largest denomination.
• Flush. The cards are all in the same suit, but the kinds of the cards do not form a consecutive sequence.
• Full House. The hand has 3 cards of one kind and 2 cards of another kind.
• Four of a Kind. The hand has 4 cards of one kind, and 1 card of another kind.
• Straight Flush. The cards are all in the same suit and the kinds form a consecutive sequence.

The Probability Density Function

Computing the probability density function is a good exercise in combinatorial probability.

The number of different poker hands is a combination problem.

The number of combinations is n! / r!(n - r)!. where N=52 and R=5

52C5 = 52! / 5!(52 - 5)! = 52! / 5!47! = 2,598,960

Next, we count the number of ways that five cards can be dealt to produce four of a kind. It requires three independent choices to produce four of a kind:

Choose the rank of the card that appears four times in the hand. A playing card can have a rank of 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king, or ace.

For four of a kind, we choose 1 rank from a set of 13 ranks. The number of ways to do this is 13C1.

Choose one rank for the fifth card.

There are 12 remaining ranks, from which we choose one. The number of ways to do this is 12C1.

Choose a suit for the fifth card. There are four suits, from which we choose one. The number of ways to do this is 4C1.

The number of ways to produce four of a kind (Num4) is equal to the product of the number of ways to make each independent choice. Therefore,

    Num4 = 13C1 * 12C1 * 4C1 = 13 * 12 * 4 = 624

Conclusion: There are 624 different ways to deal a poker hand that can be classified as four of a kind.

Finally, we compute the probability. There are 2,598,960 unique poker hands. Of those, 624 are four of a kind. Therefore, the probability of being dealt four of a kind (P4) is:

    P4 = 624 / 2,598,960 = 0.0002400960384

The probability of being dealt four of a kind is 0.0002400960384. On average, four of a kind is dealt one time in every 4,165 deals.