Question

You draw five cards at random from a standard deck of 52 playing cards. What is the probability that the hand drawn is a full house? (A full house is a hand that consists of two of one kind and three of another kind.)

Answer 1

Answer:

The probability that the hand drawn is a full house is 0.00144.

Step-by-step explanation:

In a full house we have a hand that consists of two of one kind and three of another kind, i.e 5 cards are selected.

The number of ways of selecting 5 cards from 52 cards is:

                          ${52\choose 5} = \frac{52!}{5!(52-5)!} \\=\frac{52!}{5!\times47!} \\=2598960$

In a deck of 52 cards there are 13 kind of cards, namely{K, Q, J, A, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

Two kinds can be selected in, ${13\choose 2}=\frac{13!}{2!\times(13-2)!} =\frac{13!}{2!\times11!} =78$ ways

One of the two kinds can be selected for 3 cards combination in ${2\choose 1} = 2$ ways.

There are 4 cards of each kind.

So 3 cards combination can be selected from any of the two kinds in ${4\choose 3} =\frac{4!}{3!(4-3)!} =4$ ways.

And 2 cards combination can be selected from any of the two kinds in ${4\choose 2} =\frac{4!}{2!(4-2)!} =6$ ways.

Thus, total number of ways to select a full house is:

${13\choose 2}\times{2\choose 1}\times{4\choose 3}\times{4\choose 2}\\=78\times2\times4\times6\\=3744$

The probability that the hand drawn is a full house is:

$\frac{Number\ of\ ways\ of\ Drawing\ a\ Full\ house)}{Number\ of\ ways\ of\ Selecting\ 5\ cards } =\frac{3744}{2598960} =0.00144$

Thus, the probability of playing a full house is 0.00144.