Royal Flush Percentages

28 June 2020

By John Grochowski

QUESTION: I’ve been practicing video poker on my laptop, and I looked at some statistics. They say that there are 2,598,960 possible hands. Only four are royal flushes, so 1 in 649.740.

I get that royals come up a lot more often because of the draw, 1 in 40,000 or so. What I’m wondering is if the 2,598,960 are unique hands. It sure seems like I get something like 3 of clubs, 4 of diamonds, 7 of clubs, 9 of spades, 10 of hearts a lot more than a royal flush.

If you designated something like that as the big hand, would it be as rare as a royal?

ANSWER: Yes, each hand in that 2,598,960 total is unique. Only one of those hands is 3 of clubs, 4 of diamonds, 7 of clubs, 9 of spades and 10 of hearts.

I’d point out that card order doesn’t matter in poker hands. If the order is 9 of spades, 10 of hearts, 3 of clubs, 7 of clubs and 4 of diamonds, it’s still the same hand as the one you listed and is counted in the total only once.

There are a lot more hands with limited winning possibilities than there are sure things. More than half the total – 1,302,540 hands – have no pair or better. That’s not even differentiating between high and low pairs. A pair of 2s is a better start than more than half of the initial deals.

Would the hand you designated be as rare as a royal? It would be as rare as a royal in any given suit. With 3 of clubs, 4 of diamonds, 7 of clubs, 9 of spades and 10 of hearts, you have a hand as rare as Ace-King-Queen-Jack-10 of spades. It just doesn’t pay as well.

There was once a video poker game that enabled players to designate their own jackpot hand. I don’t remember who made it and I’m fuzzy on some of the details of how it worked, but you could push buttons to designate card and suit combinations to bring the 4,000-coin payoff that usually comes with royals.

I played it, and saw others designating odd combinations that normally would be non-winning hands. I quickly came to the conclusion that the best possible combinations to designate were the standard royal flushes. Royals sometimes grow out of normal strategy of holding high cards, while playing for others involves many few-win strategies.


QUESTION: I told some other guys what you wrote about the blackjack house edge being higher with more decks because there are fewer blackjacks and because you get 10s less often on double downs. They said that wasn’t it at all, that it had to do with when you played and players going before the dealer.

ANSWER: It sounds like there’s some confusion or miscommunication.

No matter how many decks, the house edge derives from players finishing their hands before the dealer. The house wins when you and the dealer both bust, and that’s the house’s only edge.

That’s as true with one deck as with two or more.

The difference in house edge between one-deck multi-deck games is a separate issue. The difference between those games is in the frequency of blackjacks and the results on double downs.

Each card dealt out of one deck has a greater impact on the composition of the remaining deck than each card dealt from two decks, six decks or any other number. That leads to different frequencies of blackjacks and of 10s on double downs, and is enough to make a difference in the house edge.