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The frequency of the royal flush

The frequency of the royal flush

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Nearly all video poker games — and all of the most popular ones — start with five-card draw. And even though the base game is the same, the frequency of hands differs.

That’s something I’ve explained often to players who wonder why royal flushes come up more often on one game than another. Given optimal strategy, you’ll see a royal an average of once per 40,391 hands in 9-6 Jacks or Better but only once per 48,035 in 9-7-5 Double Bonus Poker.

The reason is that our drawing strategy differs according to the pay table.

That applies not only to royal flushes, but to all hands. Your strategy affects hand frequency.

Let’s look at a two examples to see how that works:

Ace, Queen, 10 and 6 of diamonds; 7 of spades: Which should you hold, all four diamonds to maximize chances at a flush or just A-Q-10 to leave a chance at a royal?

It depends on the pay able. In 9-7-5 Double Bonus Poker, the better play is to hold all four diamonds for an average return of 7.34 coins per five wagered that bests the 6.54 on A-Q-10. In 9-6 Jacks or Better, we go for the royal instead with average returns of 6.43 coins on A-Q-10 and 6.38 on all four diamonds.

That affects frequencies of paying hands. The Double Bonus hold means you’ll never draw a royal on that hand, while in Jacks or Better you’ll hit the royal jackpot once per 1,081 times you make the two-card draw. The trade off is that in Double Bonus, you’ll complete a flush once per 5.22 hands, while in Jacks or Better it’s only once per 30.89.

Other hands are affected, too, but you get the idea. The way we play that hand is part of why we draw fewer royals in 9-7-5 Double Bonus.

2 of spades, 4 of diamonds, 6 of clubs, 7 of spades, 8 of hearts: Let’s look at this one for three games: 9-6 Jacks or Better, 9-6 Double Double Bonus Poker and 9-7-5 Double Bonus Poker.

Jacks or Better and Double Double Bonus Poker pay the same 4-for-1 on straights, but have different strategies for this possible inside straight draw. In JB, the best play is to redraw for a 1.80 average return vs. 1.70 when holding 4-6-7-8. in DDB, the average remains at 1.70 on a one-card draw for an inside straight, but the value of the redraw is reduced to 1.62.

Why? Jacks or Better pays 2-for-1 on two pairs while DDB pays only 1-for-1. That enhances the value of the redraw in JB and depresses it in DDB.

Double Bonus gives us extra incentive to make the inside straight draw by paying 5-for-1 on straights. The value of a redraw remains low at 1.64 coins per five wagered while drawing to the inside straight brings a 2;13-coin average, higher than any play on this hand on the other games.

In DDB and DDB, the one-card draw brings a straight an average once per 11.75 hands. In JB, where we discard all five cards, then draw a straight once per 255.61 hands.

The strategy difference factors into the frequency of straights overall. In JB, optimal play for the full game brings a straight once per 89.05 coins. That rises to once per 79.33 in DDB and once per 66.57 in DB.

The cards we’re dealt and draw remain random results. The machine isn’t doing anything to change card frequency. We’re the ones changing frequency of winning hands, and we’re doing it through our draw decisions.

Look for John Grochowski on Facebook and Twitter (@GrochowskiJ).

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