Question

Five cards are drawn from an ordinary 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 required probability is 0.00144 or 0.144%.

Explanation:

Consider the provided information.

Five cards are drawn from an ordinary deck of 52 playing cards.

A full house is a hand that consists of two of one kind and three of another kind.

The total number of ways to draw 5 cards are:
`^(52)C_5=(52!)/(5!47!)`

Now we want two of one kind and three of another.

Let the hand has the pattern AAABB, where A and B are from distinct kinds. The number of such hands are:

`^(13)C_1*^(4)C_3*^(12)C_1*^(4)C_2=(13!)/(12!)*(4!)/(3!)*(12!)/(11!)*(4!)/(2!2!)`

Thus, the required probability is:

`(^(13)C_1*^(4)C_3*^(12)C_1*^(4)C_2)/(^(52)C_5)=((13!)/(12!)*(4!)/(3!)*(12!)/(11!)*(4!)/(2!2!))/((52!)/(5!47!))`

`=(3744)/(2598960)\\\\\approx0.00144`

Hence, the required probability is 0.00144 or 0.144%.