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# Good Poker Hands Are Tougher To Get Than You May Think

13 August 1996

Most "live" casino poker is played with seven-card hands. Video poker, in contrast, is usually based on five-card draw. And, Caribbean Stud and Let It Ride - two new table table games apt to become casino staples - also use five cards.

Video games, Caribbean Stud, and Let It Ride differ from live poker because they lack adversarial interactions. There's no bluffing or deducing opponents' hands from their demeanors or betting patterns. However, players' chances of winning are improved by skill in determining what to hold or discard at video poker, whether to bet or fold in Caribbean Stud, and how to manage the optional bets at Let It Ride.

You can find optimal strategies for each of these games. And, unless you naively believe that gambling is all luck, you'll have familiarized yourself with at least the key precepts before risking your hard-earned money.

But where do the strategies originate? As in all casino games, they're based on the probabilities of achieving particular winning results and the corresponding payoffs.

The probability of any class of poker hand - like a pair or a straight - is the number of possible hands divided into the total ways the specified class of hand can be formed. "Combinatorial mathematics" can be used to find that a 52-card deck generates 2,598,960 unique five-card hands, of which 1,098,240 are single pairs. Dividing these figures yields 0.4226 - 42.26 percent - as the probability of any pair. Similar math gives the probabilities in Table I for the hands of most interest.

To see how probability and payout lead jointly to rules for optimum play, consider the Caribbean Stud jackpot. If you're unfamiliar with the game, players can make optional \$1 side bets before the deal. A flush or better wins, paying the amount in the second column of Table II. The probability of each winner is shown in the third column. The "expected value" of the hand, the payout times the probability, is given in the fourth column.

Add the expected values for all winning hands to get the expected value for the side bet. The sum is \$0.36 + 0.0000029xJP. When the expected value equals a dollar, the amount wagered, the bet becomes "fair" and gain balances risk. If you do the algebra, you'll find 0.36 + 0.0000029xJP = 1 when JP = \$220,689.65. Lesser jackpots shift the advantage to the house, greater amounts tilt the edge to the player. So, for optimum play, make the side wager only when the progressive jackpot is \$220,689.65 or more.

That's the math. Human nature is a different story. For instance, almost every Caribbean Stud player makes the side wager - even when jackpots below about \$50,000 drag the expected value of the \$1 bet under \$0.50. It has to do with utility theory. I won't explore this topic now, except to note that it explains why solid citizens overvalue longshots offering low probability of high gain and favor situations pitting large chances of easily-afforded losses against small chances of otherwise unattainable wins.

Sumner A Ingmark, the Longfellow of the longshot, thought of it this way.

I don't need a scholar
To help bet my dollar,
In pools where the prizes
Reach staggering sizes.

 Table I Probabilities of Various Poker Hands hand probability (percent ) One pair. 42.26 Pair of 10s or better 16.25 Pair of Jacks or better 13.00 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 straight flush 0.0014 Royal 0.00015

 Table II Payouts for \$1 Side Bet on Caribbean Stud Jackpot Winning hand payout probability (percent) expected value Flush \$50 0.20 \$0.10 Full house \$100 0.14 \$0.14 Four of a kind \$500 0.024 \$0.12 Straight flush 10% of prog JP 0.0014 0.0000014xJP Royal prog JP 0.00015 0.0000015xJP

Table II
Payouts for \$1 Side Bet on Caribbean Stud Jackpot

Winning hand payout probability expected
(percent) value
Flush \$50 0.20 \$0.10
Full house \$100 0.14 \$0.14
Four of a kind \$500 0.024 \$0.12
Straight flush 10% of prog JP 0.0014 0.0000014xJP
Royal prog JP 0.00015 0.0000015xJP

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Best of Alan Krigman
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.