Probability of Royal Flush ·

23 Jun 2024, 00:00

In a standard 5-card poker hand for 1 deck:

Calculate P(Royal Flush)

Royal Flush Hands

Calculate Total 5 Card hands

Total Hands =	52!
	(52-5)! * 5!

Total Hands =	52!
	47! * 5!

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

Cancelling the 47!, we get:

Total Hands =	311,875,200
	120

Total Hands = 2,598,960

Build Probability

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

P(Royal Flush) =	Royal Flushes
	Total Hands

P(Royal Flush) =	4
	649,740

Reduce top and bottom by 4

GCF = Greatest Common Factor

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

GCF for 1 and 649740 = 4

P(Royal Flush) =	1
	649,740

Final Answer

Decimal probability = 1.5391E-6

What is the Answer?

Decimal probability = 1.5391E-6

How does the 5 Card Poker Hand Calculator work?

Free 5 Card Poker Hand Calculator - Calculates and details probabilities of the 10 different types of poker hands given 1 player and 1 deck of cards.
This calculator has 1 input.

What 1 formula is used for the 5 Card Poker Hand Calculator?

Total Possible 5 Card Hands = 2,598,960

For more math formulas, check out our Formula Dossier

What 4 concepts are covered in the 5 Card Poker Hand Calculator?

5 card poker handcombinationa mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter
nPr = n!/r!(n - r)!factorialThe product of an integer and all the integers below itprobabilitythe likelihood of an event happening. This value is always between 0 and 1.
P(Event Happening) = Number of Ways the Even Can Happen / Total Number of Outcomes

Probability of Royal Flush

In a standard 5-card poker hand for 1 deck: Calculate P(Royal Flush) Choose 5 cards from 52 cards Total 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