Ask the Slot Expert: Royal flush probabilities

Question: My friend said that he once read an article that said you were more likely to get a royal flush when you're dealt three of the cards than when you're dealt four.

That doesn't make sense to me. Is he right?

Answer: I think I read the same article, many, many years ago. Your friend missed an important part of the article. The article didn't say that you were more likely to complete the royal when you were drawing to a 3-card royal than a 4-card royal.

The article said that more of your royals will have started with a 3-card royal than a 4-card royal.

It's true that you're as close as you can be to a royal when you have four of the cards. Your chances are 1 out of 47 to draw the card you need.

You're more than halfway there with three cards to a royal, and your chances are 1 out of 1,081 to draw the two cards you need.

What are the chances of being dealt a partial royal? Dan Paymar, fortunately, did the math for us on page 164 in Video Poker Optimum Play. Your chances of being dealt a 4-card royal are 1 in 2,777 and a 3-card royal 1 in 92.

Let's put it all together. You just hit a royal flush. What is the probability that you started with a 4-card royal? The probability that you're dealt a 4-card royal is 0.00036 (1/2777) and the probability that you complete the royal is 0.021 (1/47). Multiplying the two probabilities together yields 0.0000076.

Doing the same calculations for a 3-card royal (0.011 times 0.00093) gives a probability of 0.0000102.

Those numbers are difficult to compare, so let me scale them to make them easier to compare: 76 and 102. Before you hit deal, it's more likely that you'll be dealt a 3-card royal and complete it than get and complete a 4-card royal.

So, the article didn't say that completing a 4-card royal is more likely than completing a 3-card royal.

The article said that being dealt and completing a 3-card royal is more likely than being dealt and completing a 4-card royal.

Comment: Last week I mentioned some mistakes I've seen video poker players make. Another mistake I've seen players make is having royal flush fever -- that is, going for the royal flush even when it's not the best play.

A friend of mine was once dealt the 9, jack and queen of clubs and held just the jack and queen. When I asked her why she didn't hold the 9, she said that she was going for the royal.

"What about the straight flush?" I asked.

She said that the royal paid more.

That's true, but the expected value of the 3-card straight flush is higher than the 2-card royal. It's a shame to give up the chance for a royal, but going for the straight flush will win more in the long run.

Another friend once said that his goal when playing video poker is to hit a royal. That may be the highest-paying hand on the paytable, but the goal when playing video poker is to play at the highest long-term payback possible. The royals will come when they come.

Royal flush fever expressed itself in his play whenever he held a 3-card royal over a high pair on 9/6 Jacks. It hurts to give up that partial royal, but the high pair has a higher expected value and should be held.

You can go all out for the royal when you're playing in a video poker tournament. When you're playing for real, always hold the combination of cards with the highest expected value.

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