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Best of Donald Catlin
Three Card Poker Strategy6 February 2009
The popular game of Three Card Poker, invented by my good friend Derek Webb and now marketed by Shuffle Master, has a wager called Ante-Play that involves a player decision. Let me first explain the wager and then I'll discuss the proper way to play this bet.
There are three betting spots in front of the player. The top spot is labeled Pair Plus and need not concern us in this article. Below it are two spots labeled Ante and Play, respectively; these do concern us here. Play begins with the player placing a one-unit bet in the space marked Ante. The dealer then deals out three cards to the dealer and three cards to the player; these are the dealer's and player's hand, respectively. The dealer's hand is dealt face down and is unknown to the player. The player may look at his hand. After seeing his hand the player must decide whether or not he wishes to fold his hand. If so, the player loses his Ante bet. If not the player must put an additional unit bet in the space marked Play and the dealer then exposes his hand. Independent of the dealer's hand, if the player has a straight or better (see below) the player qualifies for a payout called Ante Bonus that is awarded according to a pay table. This will not concern us here since there is no strategic decision involved.
The ranking of hands in this game is as follows (highest to lowest):
The rationale for this ranking can be found in one of my other articles in the archives.
If the dealer's hand is an unsuited Q-3-2 or higher then the dealer is said to qualify. If the dealer does not qualify then the player's Play bet is returned and his Ante wager is paid even money and the bet is settled. If the dealer does qualify then the player's and dealer's hands are compared to see who has the higher ranked hand. If the dealer's hand beats the player's hand, the dealer collects the player's Ante and Play wagers. If the player's hand beats the dealer's hand, then the player wins even money on both the Ante and Play wagers. If the hands tie then the bet is a push.
So, the player's decision is whether to fold or not. Let pw, pl, and nq represent the probabilities that the dealer qualifies and the player wins, the dealer qualifies and the player loses, and the dealer doesn't qualify, respectively. Then the expected return for the player assuming he plays his hand and doesn't fold is:
exp = 2pw - 2pl + nq
If the player folds his expected return is -1. So whenever exp > -1 the correct decision is to put up the Play wager. All we need are the above probabilities.
Once the player's hand is dealt there are 18,424 three card hands that the dealer can receive from the remaining 49 cards in the deck. That's a lot of hands. So, I wrote a computer program to carry out the calculations. Let me describe it for you.
The program begins by asking you to enter a three card hand (the player's). These cards are removed from the deck and then the program cycles through each of the 18,424 possible dealer hands. If the hand is 12-3-2 or below the program records a non-qualifying hand. If not then the program compares the hand with the player's hand and records either a win, a loss, or a tie. These totals are displayed at the end of the program. Here are the relevant facts.
The probabilities are obtained by dividing the numbers in each column by the total. I find it numerically better to multiply each of the numbers in the above table by the respective payout/loss, add them up, and divide the result by the number 18,424. Here are the results:
There you have it. Any hand that is better than 12-6-3 should be played; 12-6-3 and below should be folded. See you next month.
Don Catlin can be reached at email@example.com
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