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If someone playing Poker against you had a Full House, what are the odds you could...

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)

Solutions

Expert Solution

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.

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

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


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