Question

For each of the following, compute the probability of randomly drawing the hand (given 5 cards from the full 52). Leave your answer numerically unsimplified but in “factorial form”

(a) Three of a Kind (that is: 3 cards of one denomination together with 2 cards of two other denominations)

(b) Two Pairs (pairs of 2 different denominations, together with a 5th card of another denomination)

Answer 1

Number of ways of selecting 5 cards out of 52 cards is

(a)

Number of ways of selecting 1 denominations out of 13 is C(13,1). Number of ways of selecting 3 cards out of 4 cards of selected denomination is C(4,3). And then select two denominations out of remaining 12 denominations is C(12,2) and then 1 card from each selected denominations is C(4,1)C(4,1). So number of ways are there to draw a 5 card poker hand that contains 3 a kind is

C(13,1)C(4,3)C(12,2)C(4,1)C(4,1) = 54912 ways

So required probability is

(b)

There are total 13 denominations and each denomination has 4 cards. So number of ways of selecting 2 denominations and then 2 cards out of 4 is

And since we need exactly 2 pairs so remaining 1 card must come from different denomination so number of ways of selecting 1 denominations out of remaining 11 denominations and then 1 card from selected denomination is

So number of ways of selecting 2 pairs is :

So required probability is