A flush in 5 card stud poker is obtained if all 5 cards are

A flush in 5 card stud poker is obtained if all 5 cards are of the same suit, i.e., either 5 hearts, 5 diamonds, 5 spades, or 5 clubs. If you are dealt 5 cards, what is the probability of getting a flush?
Please write your answer as a fraction, and don\'t reduce

Solution

There are total 52 cards and we have to draw 5 cards from 52.

Possibilities for 5 cards = (52C5) where, C is used for combination.

= 52! / 5!*(52-5)!

= 52! / 5!*47!

Possibilities for 5 cards = 2598960

The number of hands to pick the 5 cards in one suit is (13C5).

13C5 = 13! / 5!*(13-5)! = 13! / 5!*8! = 1287

The number of hands to pick the suit for the flush is (4C1).

4C1 = 4! / 1!*(4-1)! = 4! / 1!*3! = 4

The total possibilites are 4*1287 = 5148.

And if you don\'t count straight flushes (all five cards sequentially ace is either high or low) as a flush then the number is 5148 - 40 = 5108

Probability of getting a flush = 5108 / 2598960