Question

If someone playing Poker against you had a Full House, what are the odds you could win against them? (We’ll assume that their cards are ‘still in the deck’ when it comes to calculating your own odds)

Answer 1

A full house is when someone has 3 cards of the same rank.

There are 13 distinct ranks-

A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K

In Poker, 4 of a Kind, Straight flush and a Royal flush beats a Full House

4 of a kind means the player has 4 cards of identical rank.

The probability to get 4 cards of the same rank from a deck of 52 cards-

=

=

= 3.69 x 10^-6

There are 13 ranks so the total probability would be the some of the 13 individual ranks-

13 x 3.69 x 10^-6= 4.8 x 10^-5

A straight flush is to obtain 5 cards, the player has to get 5 cards in a continuous order and all 5 cards have to be the same rank.

Possible combinations are-

(1,2,3,4,5) , ( 2,3,4,5,6) , (3,4,5,6,7) , ( 4,5,6,7,8) , ( 5,6,7,8,9) ,(6,7,8,9,10), (7,8,9,10,J) , ( 8,9,10,J,Q) , ( 9,10,J,Q,K)

The combination of ( 10,J,Q,K,A) is called a royal flush and is the highest card order in the game.

The probability to get (1,2,3,4,5)-

= 1/ 25,98,960

This is for 1 suite,

There are 4 suites

Spades, Clubs, Diamonds, and Hearts.

Therefore, there are 4 ways it can occur,

Total Probability = 1 / 649740 = 1.53 x 10^-6

Therefore, the total probability to beat full house- 1.53 x 10^-6 + 4.8 x 10^{-5 =} 4.953 x 10^-5 or 4.953 / 100000

Approximately, 5 /100000 = 1 / 20000 or 1 in 20,000

Therefore, the probability to beat a full house is 1 in 20,000