Using Three Card Poker to explain gaming analysis
January 17, 2017 3:00 AM
by Elliot Frome
Three Card Poker remains the most successful table game of all time, in terms of the number of tables out there. The game is so simple it also provides me with a great opportunity to explain many concepts of gaming analysis.
Three Card Poker utilizes a standard 52-card deck. To begin play, the player makes an Ante Wager. He is dealt three cards face down and the dealer is dealt three cards face down. The player looks at his cards and can either Fold, forfeiting his Ante, or make a Play wager equal to his Ante. If he makes the Play wager, his hand will go head to head against the dealer’s hand.
The dealer will reveal his cards. If his hand is not at least Queen High, the player’s Ante wager is paid even money and the Play wager pushes. If his is Queen High or better, then the player is paid even money on both wagers if his hand outranks the dealer’s hand. He loses both wagers if the dealer’s hand outranks his, and if they are identical the wagers push.
The critical piece of information that needs to be gathered is which hands should the player Fold and which should he Play. In theory, it is possible this decision point is not a clear cut line. In some games what seems like it should be a simple hand turns more complex. But, in Three Card Poker it turns out to be rather simple. If the player’s hand is Q-6-4 or better, he should Play. If it is Q-6-3 or less, he should Fold.
How do I know this and why? Let’s start with the why. The player has already made a one-unit wager. He must now decide if he is better off making another one-unit wager and playing the hand out or simply forfeiting the first unit. If he Folds, the average loss per hand is one unit. So, it makes sense he should Play any hand that results in less than an average of a one-unit loss. Because of the qualifying rule, the formula winds up looking something like this: Probability of winning hand with dealer qualifying x 4 + prob. of winning hand with dealer not qualifying x 3 + prob. of tie x 2.
To solve this equation for any given player hand we need to know the probability of winning (with the dealer qualifying and not qualifying) and of tying. With a single deck of cards, there 22,100 possible three-card hands that can be dealt first. Once three cards are removed, there are 18,424 possible additional three-card hands. So, for each of the 22,100 unique player hands, there are 18,424 dealer hands.
So, for any given three-card player hand, I can have a computer program deal out the remaining 18,424 dealer hands and determine how many result in a win (with and without qualifying), loss and tie. I now have my probabilities. Now I can find where my decision point is via trial and error.
Perhaps I start by running the lowest possible Pair Hand (2-2-3) and find this is clearly a Play. Then I try the lowest possible Ace High hand (A-2-4) and find this too is worthy of a Play, so I keep going.
I can run all 22,100 player hands against all 18,424 dealer hands and find which hands should be Played and which should be Folded. Normally, once I have the strategy, I would simulate the game. But, since the total number of hands is merely, 407 million or so, I am able to create every possible hand and calculate the payback with absolute precision.
This would be similar to how I would determine the payback of Pair Plus, the bet based solely on the player’s hand. Three Card Poker is one of the few table games where the payback is absolutely precise with no reliance on simulations.
Based on our formula, we learn hands played are not necessarily winning but rather hands that are better than losing one unit on average. So, a hand that will lose 0.95 units per hand is still worth a Play wager. This is critical to realize because you will still lose a lot of hands you don’t Fold.
All the hands that are Queen High and King High are losers in the long run. It is only when you get the hands that contain an Ace or better you can expect to win money. Although, it is ironic a hand like K-Q-10 will actually win far more often than it will lose, but because of qualifying many of these hands will result in a net win of one instead of two, while all the losses will result in a loss of two units.
It is critical to understand this about these hands or you risk losing faith in the strategy. The Queen and King high hands make up a lot of our three-card hands and can string together into losses that will wreck your bankroll. But, to deny that strategy is to deny the basic math that is behind it.
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