Charlie draws 5 cards out of a deck of 52 If he gets at leas

Charlie draws 5 cards out of a deck of 52. If he gets at least three of one suit, he discards the cards not of that suit and then draws until he again has 5 cards. For example, if he gets 3 hearts, 1 club, and 1 spade, he throws the 2 non hearts away and draws 2 more. What is the probability that he will end up with 5 cards of the same suit?

Solution

Probability = (Probability of getting 3 of the same suit in 5 cards) x (probability of getting 2 of the same suit after drawing two exchanged cards)+(probability of getting 4 of the same suit in 5 cards) x (probability of getting the same suit when exchanging the last card) + probability of getting all 5 of the same suit right of the bat

= (4C1 * 13C3 * 39C2 / 52C5 ) * (10C2 / 49C2) + (4C1 * 13C4 * 39C1 / 52C5 ) * (9C1 / 49C1) + (4C1 * 13C5 / 52C5 )

= 0.012 + 7.88 * 10^-3 + 1.98 * 10^-3

= 0.02186