A poker hand contains neither kings nor aces. Find the conditional probability it is a full house.

Question

A poker hand contains neither kings nor aces. Find the conditional probability it is a full house.

Answer 1

P(No Kings AND No Aces) = (44 c 5) / (52 c 5) since we have 44 ways of getting non-king/non-ace cards.

 

P(No Kings/No Aces AND Full House) = (11 c 1)(4 c 3)(10 c 1)(4 c 2) / (52 c 5)

 

This is because we start with 11 non-king/non-ace numbers to choose from...so we pick 1.  Then, once we have that card (say it's a 7), we need 3 of them.

Then we have 10 other non-king/non-ace number to choose from...so we pick 1.  Then we need 2 of those.

 

P(Full House | No Kings/No Aces) = P(No Kings/No Aces AND Full  House) / P(No Kings/No Aces)

= (11 c 1)(4 c 3)(10 c 1)(4 c 2) / (44 c 5) = (11)(4)(10)(6) / 1,086,008 = .00234