Plenty of possibilities in Three Card Poker

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From a mathematical perspective, every game should be as easy as Three Card Poker to analyze.

It’s one of the very few table games for which we can play every possible hand combination even for the base game and not just the sidebet. In Three Card Poker, the player gets dealt three cards from a 52-card deck. There are 22,100 possible hand combinations that the player can get. In math terms, this is called “52 choose 3.” In English, this means how many different ways can you pick three items from a total of 52 items where order doesn’t matter.

Once the player has his hand, the dealer now receives his three-card hand from the remaining 49 cards. This would be “49 choose 3,” or 18,424. It does not matter if the player’s cards are dealt first or last, or even if the cards were dealt alternating to player and dealer. In the end, the total number of player/dealer combinations is 22,100 times 18,424, or 407,170,400. This may sound like a lot of hands. But for today’s computer, all these hands can be analyzed in about an hour.

For those not familiar with Three Card Poker, let’s review the basic rules. The player makes an initial wager, called the Ante. He and the dealer each receive three cards face down. The player reviews his hand and can fold, forfeiting the Ante or make a Player wager equal to his Ante. Assuming he plays, the dealer reveals his hand. If the dealer has less than a Queen High, the Ante is paid even money and the Play wager pushes.

If the dealer has a Queen High or better, the hand “qualifies” and the player will win even money on both wager if his hand outranks the dealer. Or he will lose both wagers if the dealer’s hand outranks the player’s hand.

So when would a player want to play vs. fold? If he folds, he will lose his wager, which we’ll call “one unit.” If he plays, he will wager two units in total (the Ante and the Play) but he will have an opportunity to win some of that back. If the dealer doesn’t qualify, he will win back a total of three units.

If the dealer qualifies and the player wins, he will win back four units. If the dealer qualifies and the player loses, he will win zero units. So the player must expect to win back an average of at least one unit in order to make his net loss a total of 1 unit or less. This would make it better than folding. 

This is exactly what our computer program will be looking for. It will start with one of the player’s possible hands and simulate each of the 18,424 possible dealer hands. Remember, that the player must make his decision before seeing any of these dealer hands. So, he either folds for all of the 18,424 hands or he plays all of them. If he folds for all of them, he will lose 18,424 units. If he plays all of them, he will wager a total of 36,848 units.

The question becomes will he win back at 18,424 for a net loss of less than 18,424 or not? If he does, then the hand should be played. If his net loss would be more than 18,424 he would be better folding. 

What if the player is dealt a pair of 3’s and a deuce? Our computer program shows us that the player will win with the dealer qualifying 8,567 times, the dealer will not qualify 5,224 times and the he will push three times. The rest of the hands are losers.

When we do the math we get (8,567 x 4) + (5,224 x 3) + (3 x 2) = 49,946. The player will wager 36,848 units and win back nearly 50,000 for a net win.

The computer program will do this for all 22,100 hands. What we find when it has completed its work is that if the player has a hand of Ace High or better, that it is a net winner for him in the long run. If the hand has a King High, then it is a net loser, but the player will lose less by playing than by folding.

The critical hand is Q-6-4. At this point, the player can expect to lose “only” about 18,300 units, which is a little less than if he folded. At Q-6-3, he will lose about 18,471 units, on average and be better off folding.

The computer has done all the hard work for us. This is how the strategy of play on Q-6-4 or better was created for Three Card Poker. The computer program also tallied all of the units won and lost while playing all 407 million plus hands while using this strategy. This is how we know that the payback of Ante/Play for Three Card Poker is 97.98 percent when we include the standard Ante bonuses.

Utilizing any other strategy can have only one impact in the long run and that is to return less to the player.

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About the Author

Elliot Frome

Elliot Frome’s roots run deep into gaming theory and analysis. His father, Lenny, was a pioneer in developing video poker strategy in the 1980s and is credited with raising its popularity to dizzying heights. Elliot is a second generation gaming author and analyst with nearly 20 years of programming experience.

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