Probability of Royal Flush




Calculate the probability of a Royal Flush:
Ace of ClubsKing of ClubsQueen of ClubsJack of Clubs10 of Clubs
Ace of DiamondsKing of DiamondsQueen of DiamondsJack of Diamonds10 of Diamonds
Ace of HeartsKing of HeartsQueen of HeartsJack of Hearts10 of Hearts
Ace of SpadesKing of SpadesQueen of SpadesJack of Spades10 of Spades

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 Royal Flushes = Possible ways to get 10,J,Q,K,A all suited
Possible Royal Flushes = 1 way * 4 possible suits (Spade,Heart,Diamond,Club) = 4
Probability(Royal Flush) =Possible Royal Flushes
Total Possible 5 Card Hands

Probability(Royal Flush) =4
2,598,960

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

In decimal format, this probability is equal to approximately 1.5391E-6