Question

In poker, there is a 52 card deck with 4 cards each of each of 13 face values. A full house is a hand of 5 cards with 3 of one face value, and 2 of another. What is the probability that a random poker hand is a full house? You can leave your answer in terms of bionomial co-efficients and similar factors, but please explain each term.

Answer 1

A full house card contains three cards of one face value and two cards of different card.

Since order of the cards is not important, combinations is appropriate to proceed.

Here, total possible outcomes is ⁵²C₅.

First, one face card has to be chosen from 13 face cards. That is, ¹³C₁.

In a 52 card deck, there are 4 cards of each face value. From this, three cards of the same face value have to be chosen. That is, ⁴C₃.

Similarly, another face card has to be chosen from the remaining 12 face cards (as already one face value has been chosen). That is, ¹²C₁.

From 4 cards of face value, 2 have to be chosen. That is, ⁴C₂.

The probability that a random poker hand is a full house is calculated as follows: