The purpose of the blog is for understanding the game of Texas Holdem mathematically. I do not intend to talk anything about bluffing, which is obviously a big part of winning no-limit games. Nonetheless, understanding the game and the odds will definitely be helpful.
Texas Holdem is no doubt the most popular poker game around because of its quick turn around and excitement. The popularity was especially boosted about 5-7 years ago when some world series of poker games were first broadcasted on TV (ESPN). Since then “all in” has become like a fancy thing to say as the games broadcasted were usually no-limit games.
The rules of the game are simple enough to make almost everybody believe that they are the experts once they won a few times. Some people can give you exact winning percentages of one hand against another (almost as soon as they see the 2 hands) while “coin flip” becomes a slogan. (That humbles me a lot when I have no clue how they are all so quick in mind-math.)
On the other hand, if you observe the games for a while, you might find that some hands are not as good as you think. (Of course, if people don’t treasure their hands so much, you will lose a lot of opportunities to win money!)
An example of the “gold hands” is no doubt AK. Probably 75% or more of the players will bet the farm on it when they get the hand. So how good is the hand anyway? It’s winning percentage is only equivalent to pocket pair of 7’s! In other words, 8’s have a better change of winning than AK!
Out of the curiosity, I wrote a Java program to simulate the game and calculate winning percentages for each hand. The program takes 20+ hours to run about 3.2 billion games on my Toshiba A75 which has a Intel Pentium 4 CPU with a clock speed of 2.8 GHz. RAM is 1 GB. I will post my results in a winning percentage table.
1, From the deck of 52 cards, we select 2 cards as the control hand. For each control hand, there are 2450 (i.e., 50×49) possible opponent hands. We will then play the control hand against each of the possible opponent hands.
2, After we select a control hand and an opponent hand, there are 48 cards left in the deck. We now randomly pick 5 cards out of the 48 remaining cards for the community cards. In other words, there is no folding in my experiments. All games are played to the end.
3, For each control and opponent pair, I play 500 games. Using the results of the 500 games to calculate the winning percentage of the control hand. Therefore, for each control hand, the winning percentage is calculated based on 1,225,000 (2450×500) games played.
4, For margin of error, I will use kai-square to calculate later. However, for now as a quick estimate, the results should satisfy these simple facts:
(a) Same off-suited hands should have the same (very close) winning percentage. For example, 7 and 8 of different suits should have identical winning percentage theoretically. So the spreads can be used to estimate the margin of error.
(b) Same pocket pair should have the same (very close) winning percentage.