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Chapter 11: Problem 9

Full house: 3 cards of one number and 2 cards of a second number

Short Answer

Expert verified
The probability of getting a full house in a 52 card deck is approximately 0.0014, or 0.14%.

Step by step solution

01

Identify the necessary components

A full house in poker is created by having 3 cards of one rank and two cards of another rank. So, there are 2 components to calculate the number of combinations for a full house. Firstly, three cards of the same rank can be chosen in \( \binom {4}{3} \) ways, because from each rank (4 in total) 3 cards need to be picked. Similarly, the two cards of a different rank can be selected in \( \binom {4}{2} \) ways. As there are 13 ranks, the first component can happen in \( 13 \times \binom{4}{3} \) ways, and for the second component there are 12 remaining ranks, and it can happen in \( 12 \times \binom{4}{2} \) ways.
02

Calculate the total number of ways

Since each rank is independent, i.e. choosing a rank for 3 cards does not influence the choice of rank for 2 cards, the number of ways to get a full house is the product of the number of ways for each component. This results in \( [13 \times \binom{4}{3}] \times [12 \times \binom{4}{2}] = 3,744 \) ways.
03

Calculate the probability

The probability of an event is calculated by dividing the number of successful outcomes by the total number of outcomes. In a deck of 52 cards, there are \( \binom{52}{5} = 2,598,960 \) ways to select 5 cards. Therefore, the probability of getting a full house is \( \frac{3,744}{2,598,960} \) , which approximately equals 0.0014 or 0.14%.

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