Question

How many 5-card hands have at least one pair?

Answer 1

1. The 1^st card is drawn out of 52 cards. The probability of not drawing a par is 1.

2. When the 2^nd card is drawn, the probability of drawing the same numbered card is 3/51 so that the probability of not drawing a pair at this stage is 1 -3/51 = 48/51.

3. When the 3^rd card is drawn, the probability of the 3^rd card pairing up with either of the first two cards is 6/50 so that the probability of not drawing a pair at this stage is 1 -6/50 = 44/50.

4. When the 4^th card is drawn, the probability of the 4^th card pairing up with any of the first 3 cards is 9/49 so that the probability of not drawing a pair at this stage is 1 -9/49 = 40/49.

5. When the 5^th card is drawn, the probability of the 5^th card pairing up with any of the first 4 cards is 12/48 so that the probability of not drawing a pair at this stage is 1 - “12/48 = 36/48.

Thus, the probability of   a pair not being drawn in a 5-card hand is 1*48*44*40*36/51*50*49*48 = 0.507 (approximately).

Therefore, the probability that there is at least one pair in a 5-card hand is 1-0.507 = 0.493.

The number of 5 card hands that have at least one pair is [(52!)/(5!47!)]* 0.493 = 1281287(on rounding off to the nearest whole number).