My formula to figure how often we win

December 24, 2012 3:00 AM


When people find out what I do for a living, a frequent comment I hear is that I must be very good at math.

Well, I was always one of those good math students. But, the truth is that the type of math used in analyzing games is mostly algebra and combinatorial math. If math was never your strong suit, this still might intimidate you.

In reality, these are two subjects of math taught in high school to just about every student. Find some guy who knows quantum physics and he’ll blow me away with his math skills.

Algebra is the type of math that lets you solve a relatively simple equation for a single variable. If John is twice as old as his sister now and John was 7 when his sister was born, how old is John now? It won’t take much to figure out that John is now 14.

This is an overly simplistic problem. In the gambling world, it is this type of math that allows us to figure out what percent of the hands we need to expect to win to make a bet worth making.

Let’s say that you have to make an initial wager of 2 units. After getting to see some of your cards, you have to make an additional single unit wager or fold. If you win, you’ll get paid even money on all your wagers.

What percent of the hands do you have to expect to win in order to make the wager vs. fold? We set up an algebraic equation to solve this. We will be paid 6 units for each win (our original 3 units wagered plus the 3 we win). If we choose to fold, we wager 1 less unit (which in gambling is like winning one unit).

So, we have the following: 1 is less than 6p, where p is the probability of winning the hand.

When we solve this problem, we realize that if the probability of winning is greater than 1/6 (0.166667), then the hand is worth playing. This can narrow down our focus in trying to find the beacon hand(s). These are the hands that are right on the edge of our probability.

In the case of a game like Three Card Poker, it is the Q-6-4 hand. Of course, the equation for Three Card Poker is a bit more complex. This is because we are paid different amounts for hands depending on whether or not the dealer qualifies.

The dealer will not qualify about a third of the time in Three Card Poker. For simplicity sake, we’ll assume it is exactly 1/3 of the time. That means when a player wins a hand he can expect to be paid even money on both ante and play 2/3 of the time. His play wager push and ante pay even money 1/3 of the time.

Putting this into a similar equation as before and we have: 1 is less than (p x 3) x 1/3 + (p x 4) x 2/3.

When this equation is solved we find that the probability of winning must be greater than 3/11 or just over 27%. What this tells us is that any hand that has a probability of winning more than 27% of the time is worth playing.

This makes us realize that many of the hands we play are losing ones most of the time. When we look at the exact stats of the Q-6-4 hand we find that we will win this hand only 33% of the time. This is above the 27% described earlier for two reasons. One is that we used rough approximations for our qualify frequency and two, because there is some impact of one hand on the other.

Bottom line, you’re still going to lose this playable hand 2 out of 3 times. But, this is still better than Folding and forfeiting that unit.

No one is going to expect a player to know any of these formulas or to be able to calculate them on the fly. This is simply not necessary. I present them here as a means of showing that the strategy behind all casino games are based in solid mathematical principles. There is no gut feel. There is no guessing.

Card games create incredible naturally occurring random events that make each game so truly unique. Because we know that each deck has 52 cards, with 4 suits of 13 ranks, we can calculate with 100% accuracy every possible hand. Of course, now I’m delving into the other half of the math – combinatorial math. I’ll save that for another day.

I’d like to take this opportunity to wish all of you a very happy holiday and healthy new year. May it be filled with many Royal Flushes, too!

Elliot Frome is a second generation gaming analyst and author. His math credits include Ultimate Texas Hold’em, Mississippi Stud, House Money and many other games. His website is Contact Elliot at

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