Question

Q.7: What is the probability that a poker hand of 5 cards contain exactly 2 aces (hypergeometric distribution)?

Answer 1

Number of ways to take any 2 aces from 4 ones is $4!/2!2!=6$

Number of ways to take any 3 from 48 other cards is $48!/3!45!=17296$

Therefore, number of ways to take any 5 cards with exactly 2 aces is $6\cdot 17296=103776$.

Number of ways to take any 5 cards from a deck of 52 cards is $52!/5!47!=52\cdot 51\cdot 49\cdot 20=2598960$

Therefore, the probability that a poker hand of 5 cards contains exactly 2 aces equals to $103776/2598960\approx 0.04$.