# Probability of 4 of a kind

Calculate the probability of 4 of a kind:

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 4 of a kinds = Possible ways to get 4 cards of one kind and any 5th card
Possible 4 of a kinds = 13 different 4 of a kind choices * 48 remaining 5th cards = 624

 Probability of a 4 of a kind = Possible 4 of a kind hands Total Possible 5 Card Hands

 Probability(4 of a kind) = 624 2,598,960

Using our GCF Calculator, we see that 624 and 2598960 can be reduced by 624
Reducing top and bottom by 624, we get:
 Probability(4 of a kind) = 1 4,165

In decimal format, this probability is equal to approximately 0.000240096