The Royal Flush Flush!

30 August 2020

By John Grochowski

QUESTION: I’ve noticed something playing video poker. Every time I get four cards in a royal flush, I get an adrenaline rush before I hit the draw button. Really, though, it doesn’t turn into a royal all that often. My last royal was a two-card draw, and the one before that was a surprise with three cards.

I know that’s coincidence with small numbers of hands and all that, but how much better are my chances at drawing a royal holding four cards than with fewer?

ANSWER: Each time you draw to a hand that gives you a shot at a royal, there is only one possible draw that will give you a royal. If you hold suited Ace-King-Jack-10, only drawing the suited Queen will complete the royal. If you hold suited King-Queen, the only draw that will give you a royal is Ace-Jack-10 of that same suit.

That means the odds of drawing a royal are determined by the number of possible draws. And the more cards you hold, the fewer possible draws there are.

If you hold four parts of a royal and make a one-card draw, there are 47 possible draws. Your odds of completing the royal are 1 in 47. You’ll miss more often than you hit, but the rewards are so great you’d love to have that 1 in 47 shot on every hand.

If you hold three parts of a royal and draw two cards, there are 1,081 possible two card draws in which card order doesn’t matter. Your first card is 1 of 47, then your second card is 1 of 46 remaining at that point. Multiply 1/47 by 1/46 and you get 1 in 2,162. However, in building your hand King-Queen is the same as Queen-King. Card order doesn’t matter, so you divide that 2,162 in half had at 1 in 1,081.

That means a two-card draw gives you a 1 in 1,081 chance of completing the royal.

Your chances are 1 in 16,125 with a three-card draw or 1 in 178,365 with a four-card draw.

With a complete five-card redraw, royal possibilities are open in all four suits, There are 1,533,939 possible draws after you discard five cards. Four of those draws are royals, giving you a 1 in 383,484.75 chance at a redraw royal.

The more cards you need to draw, the weaker your chances at a royal. The odds against you get real steep, real fast. However, video poker plays very fast, with 800 hands an hour not difficult for experienced players. Those who play often see the long shots come through.

You didn’t ask, but the chances at a royal on the initial deal are 1 in 649,740. Odds are longer on the deal than in a redraw because there are more possible hands with 52 cards than when you’ve narrowed the deck to 47.


QUESTION: I agree slots with 40 lines and 40-cent minimums are really 40-cent games, but everybody calls them penny slots if it’s 1-cent per line. That doesn’t bother me too much. What does bother me is when they have the 40-credit button deactivated and you have to bet at least 80 cents on a 40-line game. That’s not a penny game, that’s a 2-cent game.

ANSWER: The grumpy old pedantic in me would like to label every slot machine by its minimum bet, so your game would be an 80-cent game.

But yes, if you’re going to accept labeling slots by their minimum wager per payline, then a machine that requires 2 cents per line is not a penny slot.