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Best of Alan Krigman

Gaming Guru

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Why Few Players Want Equitable Poker Payoffs

22 March 1999

Some of the new games you may have noticed, or now favor, at your friendly neighborhood casino are based on poker values. The poker paradigm offers players several benefits:

  • Most casino patrons are familiar with the central concepts.
  • Hands are complex enough to be interesting and to let bettors improve their chances by making informed decisions.
  • High- as well as low-probability results lead to variable payout schedules, allowing frequent pushes or even-money wins as well as a shot at a sporadic giant jackpot.

Along the slot aisles, permutations aplenty of video poker attest to the appeal of this approach. In the table pits, Caribbean Stud and Let It Ride are already casino staples and other poker-value games are in the offing -- a few of which are bound to endure.

Some solid citizens, mainly those who realize luck isn't the only factor separating winners from those who had a good time anyway, like to know the chances of receiving various hands and therefore of earning different amounts. The following list gives the probabilities of each class of poker value in a five-card hand, when the game presents no opportunities to improve by drawing.

hand probability (%)
no pair 50.12
one pair 42.26
two pairs 4.75
three of a kind 2.11
straight 0.39
flush 0.20
full house 0.14
four of a kind 0.024
non-royal str flush 0.0014
royal flush 0.00015

Casinos have latitude in specifying payoffs corresponding to various hands. They can proportion winnings equitably according to the probabilities of each outcome, or bias returns to overpay at some levels and underpay at others. The big bosses can also balance what they take off the top to raise or lower house edge.

The list below shows wins per $1 bet for equitable payoffs with the indicated chances. These apply to 5-card hands with no draw, independence of other player or dealer results, wins starting at one pair, and 3 percent house edge. If wins began at jacks or better rather than any pair, probability and equitable payoff at this level would be 13 percent and $0.40 per $1, not 42.26 percent and $0.12 per $1, but nothing else would change.

hand equitable win per $1 bet
one pair $0.12
two pairs 1.10
three of a kind 2.48
straight 13.42
flush 26.18
full house 37.40
four of a kind 218.15
non-royal str flush 3,739,68
royal flush 34,903.67

These payoffs can't be compared directly with those in actual games because each situation has unique twists. For instance, in Caribbean Stud, you won't make the "call" bet on most hands below one pair. And in Let It Ride, you can win multiple wagers if the first three or four cards you see are promising, but will usually lose only one unit if they are not. However, the figures do suggest that pay schedules typically tend to load the low end, at the expense of returns on hands less likely to be encountered.

Say a game pays 1-to-1 on a pair, 2-to-1 on two pair, and 1,000-to-1 on a royal -- with intermediate returns adjusted to hold house edge at 3 percent. On a royal, a winning $5 bettor would get a crummy $5,000 instead of almost $175,000. But for every one royal, players will get over 280,000 pairs paying $5 instead of $0.60 and nearly 32,000 two-pairs yielding $10 rather than $5.50. Is this the American way or a Commie conspiracy? Your view doubtless depends on whether you're among the hundreds of thousands overpaid on commonplace hands, or the one out of nearly 70,000 shorted on a rare straight flush or royal. The prominent proponent of poetic justice, Sumner A Ingmark, said it like this:

Folks who think that life is fair,
Oft have more than their fair
Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.