A fivecard poker hand dealt from a deck of 52 playing cards

A fivecard poker hand dealt from a deck of 52 playing cards is said to be a full house if it consists of three of a kind and a pair. If all the fivecard hands are equally likely, what is the probability of being dealt a full house?

Solution

The number of ways in which we can be dealt a particular full house, says three kings and two jacks is (4C₃)(4C₂). Since there are 13 ways of selecting the face value for the 3 of a kind and for each of these there are 12 ways of selecting the face value for the pair, there are altogether. m = 13.12.(4C3)(4C2) = 3744 different full house.

Also, the total number of equally likely five-card poker hands is n = (52C5) = 2598960.

The required probability is m / n = 3744 / 2598960 = 0.00144