Question

A full house in a game of card is a hand where three cards share one rank and two cards share another rank. How many ways are there to get a full-house? What is the probability of getting a full-house?

Answer 1

Total number of cases = picking 5 cards out of 52 without replacement =

52 5

Total number of cases = (52*51*50*49*48)/(5*4*3*2*1)

Total number of cases = 2598960

Total number of cases of full house = Picking any pair of two sets out of 13*Picking 2 out of 4*Picking 3 out of 4

Picking any pair of two sets out of 13 = $\binom{13}{2}$

Picking any pair of two sets out of 13 = (13*12)/(2*1)

Picking any pair of two sets out of 13 = 78

Picking 2 out of 4 = $\binom{4}{2}$

Picking 2 out of 4 = (4*3)/(2*1)

Picking 2 out of 4 = 6

Picking 3 out of 4 = $\binom{4}{3}$

Picking 3 out of 4 = 4

Total number of cases of full house = 78*6*4

Total number of cases of full house = 1872

P[ Full house ] = Total number of cases of full house / Total number of cases

P[ Full house ] = 1872/ 2598960

P[ Full house ] = 0.00072