# Winning formulafor 3-card poker

Apr 24, 2006 1:46 AM

The first time my wife sat down to play Three Card Poker, a woman sitting next to her tried to give her some advice. She told her, for the most part, she should ”˜play’ with a queen or better, but sometimes, you can beat the dealer even with a jack.

My wife and I had already reviewed the proper strategy (play any hand that is Q-6-4 or better, and fold anything else). She desperately wanted to tell the woman who I was and why her strategy was not so good, but I prefer to keep a low profile when I’m in the casino. I’ll gladly respond to questions from people, but when I play, I prefer anonymity.

Needless to say, it did not take very long (less than an hour) for the woman to lose her \$300 or so bankroll. As she finished up, she asked the pit boss for a comp to the buffet and he happily obliged.

You got the idea that this was not the first time the woman had lost this type of money in such a short period of time. Unless she doesn’t mind paying \$300 for a \$10 buffet, she’d be far better off spending \$5 on "Expert Strategy for Three Card Poker" and finding out exactly why she’s getting killed using her strategy.

As I’ve demonstrated in this column before, the Q-6-4 is the beacon hand for Three Card Poker. It is the lowest ranking hand in which the player can expect to lose less by playing than by folding. This is determined by playing every possible dealer hand against every possible player hand and tabulating the results.

There are 18,424 possible dealer hands for any given player hand. In the case of Q-6-4, the player will win back a total of 18,546 units if he wagers each of these (total wager = 36,848) for a loss of 18,302. This compares to a loss of 18,424 if he folds. So, the player winds up 122 units ahead by playing this hand.

We can look at the woman’s strategy in two parts. The first is her decision to play any hand with at least a queen instead of the full strategy of Q-6-4. This is not the part that is hurting her very much. First of all, there are not that many hands that meet this criterion. Second, even the worst possible queen high hand (Q-3-2) will result in a return of 17,191 units.

The player is still giving up 1200+ units or about 6 percent more than he needs to, and I certainly would not recommend this strategy. Still, this is not what is causing the woman to lose \$300 in an hour, playing at a \$5 table.

The problem is the part of her strategy that causes her to think that "sometimes you can beat the dealer with a jack high hand. If we look at the highest ranking jack high hand (J-10-8), we find that the player will only get back 15,831 units for a net loss of 21,017 units. This is nearly 15 percent more than if she folded.

So, what would the expected loss be if someone were to play the worst possible hand, a 2-3-5? Strangely enough, it is the exact same 21,017! Because the dealer must have a queen high or better to qualify, a player’s hand of jack high or five high is completely the same!

Her thought that you can beat the dealer with a jack is the fatal flaw in her strategy. The bottom line is that if the dealer does not qualify, it will not matter if the player has a J-10-8 or a 2-3-5. The player will "win."

Of course, the player will win one unit (the play wager pushes), while he wagered two units on an event that will occur roughly 1/3 of the time. Talk about a terrible payback!

To add to the woes is the fact that the player will wind up with a jack high or less about 30 percent of the time. Playing 30 percent of the hands at a 15 percent greater disadvantage than the player needs to will lead to lots of free buffets, if losing \$300 to earn one can be considered free.