# Probability of Flush

Calculate the probability of a Flush:

First calculate the total number of possible hands in a 52 card deck:
From a deck of 52 cards, we want the number of possible unique ways we can choose 5 cards.
Using the combinations formula 52 choose 5 shown here, we get:
 Total Possible 5 Card Hands = 52! (52-5)! * 5!

 Total Possible 5 Card Hands = 52! 47! * 5!

 Total Possible 5 Card Hands = (52 * 51 * 50 * 49 * 48) * 47! 47! * (5 * 4 * 3 * 2 * 1)

Cancelling the 47! on top and bottom we get:
 Total Possible 5 Card Hands = 311,875,200 120

Total Possible 5 Card Hands = 2,598,960

Possible flushes = Possible ways to get 5 of the same suit.
Possible flushes in one suit * 4 suits = 13! * 4/((13 - 5)! * 5!) = 5,108 ways
Total Possible flushes * 4 possible suits = (13 * 12 * 11 * 10 * 9 * 4) / (5 * 4 * 3 * 2 * 1)
Possible flushes - (36 straight . 4 Royal {Flushes}) = (617,760 / 120) - 40
Possible flushes = 5,148 - 40
Possible flushes = 5108

 Probability of a flush = Possible flushes Total Possible 5 Card Hands

 Probability(Flush) = 5,108 2,598,960

Using our GCF Calculator, we see that 5108 and 2598960 can be reduced by 4
Reducing top and bottom by 4, we get:
 Probability(Flush) = 1,277 649,740

In decimal format, this probability is equal to approximately 0.0019654015