Have More Fun Playing Texas Holdem in Five Minutes

Would you rather spend time doing arithmetic or reading your friends' faces? My brother and I made a web tool that handles the statistics and the arithmetic for you, so you can concentrate on the fun part. When it's your turn to bet, go to pokerpower.herokuapp.com and type in the cards and push Calculate. The web tool tells you your hand power and your expected winnings.

Your hand power is the likelihood, from 0 to 1, that you will beat one opponent. It is calculated by computing all the different hands that your opponent can get, taking into account the cards on the table (if any) and the ones in your hand which he cannot have and cannot draw into. Then the program computes your possible hands of 7 cards, based on the ones you have already and the ones on the table (if any). Each of your possible hands will beat some number of your opponents' hands. The program takes a weighted average of these scores to determine the likelihood that you can beat one opponent. If it's greater than 0.5, then you're more likely to win.

When playing against several opponents, you also want to know your expected winnings. The more opponents you face, the harder it is to beat all of them. A rough approximation of your chance to win is to take the Nth power of your hand power. That is, if you have a 0.5 chance to beat one opponent, you have a 0.5^3=0.125 chance to beat three opponents. At the same time, if you beat 3 opponents, you'll win three times as much as you bet. So in this example, your expected winnings would be 0.125*3x = 0.375x. In this case, since the expected winnings are less than 1x, it's not worth betting. You can always bluff, but it will be more risky when your expected winnings is less than 1.

After computing a few hand powers using the web tool, you become more accurate at guessing the power of a hand. You can get a more solid feeling of when the odds are in your favor for more than 1x expected winnings. Let's look at some of the patterns that showed up.

Step 1: Patterns in Hand Power

I used the web tool to calculate the hand power of several starting hands in Texas Holdem. Plotted on the same chart above, what patterns appear?

Most hands are clustered around 0.5. This means you don't really know yet whether it will be a good hand. If Texas Holdem used 6 or 7 shared cards instead of 5, then I think all the hands would get even closer to 0.5. It's almost like the shared cards are watering down each person's hand.

The pairs are a big exception to this rule. They stick out from about .55 to .77. It shows the power of pocket aces, that they stand out so far from the rest. But the other pairs are good, too! A pair of fours is just as good as a king and ace of the same suit, which you might have thought was a very good hand.

I used to think that two cards of the same suit was a valuable thing. But looking at the lines, the hand power only improves by about .02 when the two cards have the same suit. A flush is just not very likely at this stage.

There's some jaggedness to the left-most lines and I think we can guess what causes it. A two and a three are still decent because they can be made into a straight. A two and a seven are much harder to form into a straight.

The value of having two cards adjacent, compared to two cards far apart, seems to be about 0.06. That's about three times the value of matched suits! However, we're assuming the same high card here so the average card value has increased, too.

You can make your own plot showing different hands, or showing these same hands once some shared cards have been revealed. I like to try to explain the trends by imagining what hands could be made. Now let's look at the other number, the expected winnings.

Step 2: Patterns in Expected Winnings

If you plot your hand power on the left and the number of players (including you) on the bottom, you get the graph above. The yellow-orange line labeled "1x" is the break-even line. Suppose you have a hand power of 0.7. According to the 1x line, you can face off against up to 5 opponents (6 players total) and still have reasonable expected winnings. If you had a better hand, your chances of winning would increase and so your expected winnings would go up. Or, on the other hand, if your hand power were 0.7 in a group of 3 players, your expected winnings would be 1.4x. To me this means that I should fold more often in a game with more people. It's just hard to beat that many opponents.

Now what if some opponents have already folded? On one hand, this means you have fewer opponents to beat. But on the other hand, the people who folded probably had low hand power (meaning bad cards). The remaining opponents will be that much harder to beat. So it is probably a reasonable approximation to just count the total number of opponents, rather than the ones still in the game.

Strictly speaking, an opponent who folds also forfeits his bet. So you can play against two people and yet win more than 3x your money. But then the ratio worsens the more you put in, since you're diluting the contribution from the people who folded. For me it's a little too complicated to do the arithmetic, but I might keep it in mind during a game.

These numbers are of course no substitute for your better judgement. If the three shared cards appear of the same suit and your friend Joe goes all in, now you have to guess whether he has a flush. Does this development make the calculations useless? I would say yes and no. Yes the numbers are useless because they assume a uniform distribution of hands, whereas you know something about your opponents' hands based on what they bet. On the other hand, if that sly guy has got you stumped and you just don't know whether he has it or not, you can just play it by the numbers and ignore his aggressive betting. Either way, I hope you have more fun playing Texas Holdem by letting your computer or your phone do the arithmetic for you.

In summary:

1. When it's your turn to bet, use the tool to find your hand power and expected winnings.

2. If your expected winnings are > 1x, it is probably safe to bet. How much is up to you.

Disclaimer: Some people may get angry if you use a calculator like this in a game of poker. Please ask them for permission before using it. The web tool is intended to make poker more fun, not to give select players an unfair advantage.

Disclaimer #2: The web tool is not 100% perfect. There are a lot of combinations of poker hands, and the calculation is very complex. I have checked many parts of the code thoroughly to verify that the statistics agree with known values. There is one approximation made to handle the case of flushes, to save memory and cpu cycles. This approximation probably makes the results accurate within .01 hand power. In any case, there's no guarantee.

That said, if you do have fun and win a lot of plastic chips, I'd love to hear about it :). Maybe tell us your hand powers at each step in a round, and walk us through your betting. It's also fun to look at famous movies and calculate their hand powers. For example, in the Bond film Casino Royale, at the final hand, both Bond and Le Chiffre have really ordinary cards dealt to them (power around 0.5). They're playing together with two other people, so they have much less than 1x expected winnings. Bond's and Le Chiffre's hands do become extremely good later on, but if they followed my numbers, they would have folded! So much for "Le Chiffre" ("the number")!