Respuesta :
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:
[tex]{52\choose 5} = \frac{52!}{5!(52-5)!} \\=\frac{52!}{5!\times47!} \\=2598960[/tex]
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, [tex]{13\choose 2}=\frac{13!}{2!\times(13-2)!} =\frac{13!}{2!\times11!} =78[/tex] ways
One of the two kinds can be selected for 3 cards combination in [tex]{2\choose 1} = 2[/tex] ways.
There are 4 cards of each kind.
So 3 cards combination can be selected from any of the two kinds in [tex]{4\choose 3} =\frac{4!}{3!(4-3)!} =4[/tex] ways.
And 2 cards combination can be selected from any of the two kinds in [tex]{4\choose 2} =\frac{4!}{2!(4-2)!} =6[/tex] ways.
Thus, total number of ways to select a full house is:
[tex]{13\choose 2}\times{2\choose 1}\times{4\choose 3}\times{4\choose 2}\\=78\times2\times4\times6\\=3744[/tex]
The probability that the hand drawn is a full house is:
[tex]\frac{Number\ of\ ways\ of\ Drawing\ a\ Full\ house)}{Number\ of\ ways\ of\ Selecting\ 5\ cards } =\frac{3744}{2598960} =0.00144[/tex]
Thus, the probability of playing a full house is 0.00144.
