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What is the probability that 5 card hand contains a four-of-a-kind, where the fo...

Feb 15, 2024

What is the probability that 5 card hand contains a four-of-a-kind, where the four-of-a-kind rank is a face card?

Solution by Steps

step 1

Calculate the number of ways to get four-of-a-kind with face cards

step 2

There are 3 face cards (Jack, Queen, King) in each suit, so there are 3 ways to choose the rank for the four-of-a-kind

step 3

For each four-of-a-kind, there is 1 way to choose the four cards of the same rank

step 4

Calculate the number of ways to choose the fifth card

step 5

There are 52 cards in a deck, and after removing 4 cards of the four-of-a-kind, there are 48 cards left to choose from for the fifth card

step 6

Calculate the total number of possible 5-card hands

step 7

There are

\binom{52}{5}

ways to choose any 5 cards out of 52 without any restrictions

step 8

Calculate the probability

step 9

The probability is the number of favorable outcomes divided by the total number of possible outcomes

step 10

The probability of getting a four-of-a-kind with face cards in a 5-card hand is

\frac{3 \times 1 \times 48}{\binom{52}{5}}

step 11

Simplify the probability expression

step 12

The probability is

\frac{144}{\binom{52}{5}}

step 13

Calculate the binomial coefficient

\binom{52}{5}

step 14

\binom{52}{5} = \frac{52!}{5!(52-5)!}

step 15

Simplify the binomial coefficient

step 16

\binom{52}{5} = \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1}

step 17

Calculate the final probability

step 18

The probability is

\frac{144}{2598960}

step 19

Simplify the fraction to get the final answer

step 20

The probability is approximately

\frac{1}{18066}

Answer

The probability that a 5-card hand contains a four-of-a-kind where the four-of-a-kind rank is a face card is approximately

\frac{1}{18066}

Key Concept

Calculating the probability of a specific poker hand

Explanation

The probability is found by dividing the number of ways to get a four-of-a-kind with face cards by the total number of possible 5-card hands from a standard deck.