Besides using Monte Carlo methods to approximate area or volume, as seen in the book, we can also use it for a more “frivolous” purpose. Many online poker calculators use Monte Carlo simulation to analyse the strength of poker hands. Given a deck of 52 cards, we introduce constraints of the 2-card holding of the player, number of other players involved, as well as the number of community cards available (between 0 and 5).

The poker calculator then simulates the dealing of random hands x number of times according to the stipulated constraints (x trials). Thus, the poker calculator is able to calculate the theoretical possibility of the hand winning against random hands held by the opponents as x tends towards infinity. The interesting issue is that while there are theoretically millions of final 7-card holdings (133,784,560 to be exact. The result of (52,7)), and we ideally want x to be a large multiple of this number of holdings, this is computationally time-consuming, especially for the situation where we have few community cards and multiple opponents.

Let’s look at an example to see why. A full iteration of all possible outcomes for hole-cards of Ace-King suited against random cards held by another opponent takes over 299 seconds with no community cards. This is because the code is evaluating over 2 billion possible hand match-ups. The exact probability of winning is 67.0446323%. The time taken decreases with increasing number of constraints. With 3 cards exposed, there are only about 1 million possible hand match-ups to be calculated, taking only a fraction of a second.

Luckily for us, poker is more forgiving with errors than academia is. Given that poker incorporates the element of bluffing (he could be raising with absolutely nothing), which impacts the calculation of the probabilities, and that a 0.5% change doesn’t really influence the decision to check, raise or fold that much, we see that approximate values of the probabilities are more than sufficient. If we calculated the probability of Ace-King suited winning against a random hand using a Monte Carlo simulation of 1000 trials, we get a value of 65.65%, which is less than 2% away from the actual answer. If we repeated it for 150,000 trials, we get a value of 67.01%, which is only 0.03% away from the actual answer. This only requires a computational time of 0.1 seconds.

Hence, we observe that Monte Carlo simulations are an effective way of estimating the probabilities of winning in poker, because the relatively large margin of error allows us to calculate very usable probabilities even with a small number of trials.

References:
http://pokerforprogrammers.blogspot.com/ provides poker algorithms for programmers