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

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

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} = (52!)/(5!(52-5)!) \\=(52!)/(5!*47!) \\=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}=(13!)/(2!*(13-2)!) =(13!)/(2!*11!) =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} =(4!)/(3!(4-3)!) =4` ways.

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

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

`{13\choose 2}*{2\choose 1}*{4\choose 3}*{4\choose 2}\\=78*2*4*6\\=3744`

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

`(Number\ of\ ways\ of\ Drawing\ a\ Full\ house))/(Number\ of\ ways\ of\ Selecting\ 5\ cards ) =(3744)/(2598960) =0.00144`

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