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# How Easy Is It to Win What at Three-Card Poker?

13 November 2002

Three-card poker has gained a loyal following in a relatively short time span. In part, because solid citizens think it's easy to win big. The most common way the game is being spread requires equal bets on "pair plus" and "ante." Players each receive three cards, which they grade according to poker value. In decreasing order of merit, hands are straight flushes, triplets, straights, flushes, pairs, and singletons. Within each category, rankings coincide with standard poker conventions, starting at the top with aces and kings and proceeding downward to threes and twos.

Players have two options during the course of the action.

They can fold, sacrificing both bets and dropping out of the round. Optimum strategy for three-card poker is to muck a "low singleton" (queen-six-three or worse).

They can place a third wager, on "play," of the same amount as the other two - for three equal units in all. Optimum strategy is to bet on "play" with a "high singleton" (queen-six-four or better). Rounds are then resolved in up to four different ways.

1) Regardless of the dealer's hand, the bet on "pair plus" pays 40-to-1 for straight flushes, 30-to-1 for triplets, 6-to-1 for straights, 4-to-1 for flushes, and 1-to-1 for pairs. It loses on anything less than a pair.
2) If the dealer "qualifies" with a queen-high or better, "ante" and "play" both win 1-to-1, push, or lose when the player has a higher, equal, or lower hand, respectively.
3) If the dealer has less than queen-high, and therefore does not "qualify," "ante" pays 1-to-1 and "play" pushes.
4) Regardless of the dealer's hand, the bet on "ante" is paid a bonus of 5-to-1 on straight flushes, 4-to-1 on triplets, and 1-to-1 on straights.

Wins and losses when "pair plus" and "ante" are both mandatory can best be stated in terms of combined net with a \$1 base bet on each element - \$2 at risk when the round begins. If you wager \$10 on each, multiply the base values by 10, and so forth.

The greatest possible win is \$47 times the base. This occurs when a player has a straight flush, bets on "play," and beats a qualifying dealer. The apportionment is \$40 for the "pair plus," \$5 for the "ante bonus," and \$1 each for "ante" and "play." The worst loss is \$3 times the base. This happens when a player bets on "play" with a singleton and gets zonked by a qualifying dealer. All three bets then lose.

The accompanying table gives a complete list of wins and losses, with associated probabilities, for optimum play. The figures show that net wins are expected in about 32.1 percent of all rounds, losses in 54.7 percent, and pushes in the remaining 13.2 percent.

Three-card poker wins and losses following optimum strategy

 amount probability hand decision dealer result +47 0.15154% SF play Q player wins +46 0.06538% SF play NQ -- +45 0.00004% SF play Q player ties +43 0.00024% SF play Q player loses +36 0.16351% 3K play Q player wins +35 0.07097% 3K play NQ -- +32 0.00081% 3K play Q player loses +9 2.20434% ST play Q player wins +8 0.98262% ST play NQ -- +7 0.00450% ST play Q player ties +6 3.14205% FL play Q player wins +5 1.57326% ST play Q player loses FL play NQ -- +4 0.00079% FL play Q player ties +3 8.88294% PR play Q player wins +2 5.44905% FL play Q player loses PR play NQ -- +1 9.36825% PR play Q player ties HS play Q player wins 0 13.23175% HS play NQ -- -1 2.97378% PR play Q player loses HS play Q player ties -2 32.57919% LS fold -3 19.15483% HS play Q player loses

(SF: straight flush, 3K: triplets, ST: straight, FL: flush, PR: pair, HS: high singleton, LS: low singleton, Q: qualifies, NQ: doesn't qualify)

Analysis accordingly belies the belief that three-card poker is easy to win. Can the math be wrong? Does what comes out of the computer differ from what spews from the shuffling machine? Maybe the faithful haven't counted their cash carefully. Or considered the caveat of the acclaimed coupleteer, Sumner A Ingmark:

Contorting a theory to foregone conclusion,
Is many a gambler's expensive delusion.

Recent Articles
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.