Question

You have a deck of 52 playing cards you are dealt 4 cards. What is the probability you are dealt exactly one pair?

Answer 1

4 cards with exactly one pair:

Pattern is:

AABC

where A, B and C are from the distinct kinds of cards: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10,Jack, Queen and King.

There are 13 kinds.

Each kind has 4 suits: Club, Diamond, Spade and Heart.

For one pair AA:

Number of choices for kind:

1 kind can be chosen from 13 kinds in:

Once 1 kind is chosen for AA, one suit for each of the two A can be selected from 4 suits in:

Once kind is selected for AA, 2 kinds for B and C (one kind for each) can be selected from remaining 12 kinds in:

1 suit can be selected from 4 suits for C in:

1 suit can be selected from 4 suits for D in:

Thus Total   number of ways of exactly 1 pair = 13 X 6 X 66 X 4 X 4 = 82368

Total number of selecting 4 cards from 52 cards is given by:

So,

Probability of exactly 1 pair = 82368/270725 = 0.3042

So,

Answer is:

0.3042