Royal flush percentages

8 July 2018

By John Grochowski

Video poker players have long been told their chances of drawing a royal flush on any given hand are about 1 in 40,000.

The precise odds depend on game and playing strategy. In 9/6 Jacks or Better, your royal chances are about 1 in 40,391, while in 9/6 Double Double Bonus Poker they're 1 in 40,799 and in 9/7/5 Double Bonus Poker they're 1 in 48,035.

Among the strategy reasons royals occur less frequently in DDB than in JoB is that we draw to more inside straight in DDB, eliminating the slim chance at a royal on the redraw. And one reason royals come even less frequently in DB is the 7-for-1 payoff on flushes lead us to hold three cards of the same suit when we'd be holding just the high cards in other games.

In any game, the 1 in 40,000 or so figure is an estimate. For specific hands, the odds of drawing a royal depend on what we see on the deal.

Initial deal: Your chances of getting a royal on the first five cards are the same as in five-card stud poker games: 1 in 649,740: There are 2,598,960 possible five-card combinations in which card order doesn't matter. Four of those are royal flushes, so the chances of a royal on the deal are 1 in 649,740.

Complete redraw: If you receive no royal cards on the deal and make a complete redraw, the elimination of the five cards in your initial hand leave 1,533,939 possible combinations from the remaining 47 cards. Divide that by the four possible royals and you have a 1 in 383,484.75 chance of a royal on a complete redraw.

At times, your discards will include a 10. When that happens, there are still 1,533,939 possible redraw combos, but only three are royals, making your chance at a royal on redraw 1 in 511,313.

Hold one card to a royal, draw four: There are 178,365 possible draws, but only one possible royal — your royal chances are committed to the suit of the card you're holding. That leaves your chance of a royal at that point at 1 in 178,365.

Hold two cards to a royal, draw three: At this point, your chances of drawing a royal become better than the 1 in 40,000 or so average.

When you hold two cards, there are 16,215 possible three-card draws in which card order doesn't matter. One of those brings your other three royal cards in the suit you need. Your chances of a royal with a three-card draw are 1 in 16,215.

Hold three cards to a royal, draw two: We're zeroing in on the goal. There are 1,081 possible two-card draws, and one is the combo you need. So you have a 1 in 1,081 shot at a royal. That's still a long shot, but it beats the 1 in 649,740 on the deal and at least brings you within dreaming distance.

Hold four cards to a royal, draw one: Now we're talking. You've already seen five of 52 cards. There are 47 available for the draw. One is the card you need, giving you a 1 in 47 shot. That's good enough that we break up winning hands such as high pairs, straights and flushes to go for the gold, though we keep straight flushes intact in non-wild card games.

Until a royal is in hand, we're never a favorite to get there. Even our best shot before the draw is only 1 in 47. But there are never any sure things in any game.

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