Calculate the probability of a Royal 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 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
