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

a) P( Three of a Kind )= P(select a denomiantion from 13 , pick 3 from 4 of its cards and then select denomination from remaining 12 and select 1 card from 4 of each)

=(13C1)*(4C3)*(12C2)*(4C1)*(4C1)/(52C5)

(13!/(12!*1!))*(4!/(3!*1!))*(12!/(2!*10!))*(4!/(3!*1!))*(4!/(3!*1!))/(52!/(5!*47!))

b)

P(two pairs) =P(select 2 denomination from 13 and select 2 cards from 4 of each denomination , then select 1 denomination of remaining 11 and select 1 from 4 of its card)

=(13C2)*(4C2)*(4C2)*(11C1)*(4C1)/(52C5)

=(13!/(2!*11!))*(4!/(2!*2!))*(4!/(2!*2!))*(11!/(1!*10!))*(4!/(1!*3!))/(52!/(5!*47!))