Estimating the outcome of a Texas hold’em game using Monte Carlo simulation
Card games were always fascinating for both players and mathematicians due to the factor of uncertainty or, in less formal words, luck. For many years, they were also the moving force of relatively modern areas of mathematics such as probability theory and statistics.
What is a Monte Carlo simulation?
In layman’s terms, Monte Carlo simulation is a technique for estimating a variable which depends on one or more random factors. This is achieved by repeated random sampling from a source of data (i.e a known distribution, a set of possible poker hands or from multiple tosses of a coin etc).
As you can easily understand, Monte Carlo, as probably the name suggest (no pun intended), is the perfect tool for estimating the outcome of various games that contain a factor of uncertainty.
Texas Hold’em: Let’s remember the game
Texas hold ’em is one of the most popular variants of poker and it is played according to the following rules.
At the start of a round, players receive two down cards as their personal hand (holecards), after which there is a round of betting. Then, three board cards are turned in the middle of the table simultaneously (called the flop) and then another round of betting occurs. The next two board cards are turned one at a time (the first one is called “The Turn” and the second one is called “The River”), with a round of betting after each card.
The board cards are community cards (common within all players), and a player can use any five-card combination from among the board and personal cards. A player can even use all of the board cards and no personal cards to form a hand (“play the board”).
Finally, using the best five-card combination of their hole cards and the community cards, the players show their hands, with the bettor or last raiser showing first. The player with the five-card combination of highest strength wins the pot.
Below, you can see a chart showing the strength ranking (from highest to lowest) of each valid Texas Hold’em poker combination.