The Importance of Probability and Bayes’ Theorem in Poker
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Every competition has its foundations in math and logic: football, baseball, chess, tennis, etc. all use tactics and plays to target the opponent’s weakest points. One of the most important uses of statistics for a sport is poker. We may just watch poker for entertainment, but for the majority of players we observe, playing poker is their professional careers, and they want to make the most informed decisions as possible. When everyday civilians play poker, we mainly bet on how we “feel” about a hand, and in many cases, we leave it up to luck and emotions to determine our outcome. Meanwhile, professional players are leaving it up to statistics, performing countless mental calculations in order to have the statistically best probability of winning. The article attached gives some insight into the applications of probability in poker.
In some instances, poker players can “read” other people’s emotional responses and determine whether they have a good hand. However, these instances are very few and far between in professional matches, since most professionals can keep a steady face and not display any outwardly sign of vulnerability. Thus, players need to depend on something much more reliable. As such, probability plays an enormous role in deciding whether a poker player should bluff, call, raise, fold, or bet. For example, if you are dealt five cards, the probability of having a four of a kind would be around 0.00024. Only two other hands top a four of a kind, and the probability of having them is .000015. Thus, the chance of any other player beating you would be extremely small, and you could be confident with betting a larger amount of money.
Furthermore, each player has to take into account all of his opponents’ actions as well and come up with updated probabilities conditioned on their past decisions. For example, if a player decides to raise the pot by $100, what is the probability that he has a worse hand? This could be modeled in a conditional probability P(A|B), where A is the event he has a worse hand and B is the event that he raises. As we’ve learned in class, P(A|B) = P(A and B) / P(B). There are certain ways to determine the probability that he raises from previous research and observations. However, it can be hard to determine P(A and B) since it is tough to observe two events at the same time in poker. Instead, we need to rely on Bayes theorem, which states P(A|B) = P(B|A)*P(A)/P(B). Bayes’ Theorem is useful in this circumstance because we can observe the probability he has a “bad hand,” or a hand that isn’t as good as yours, by statistics, and we can also tell the probability that he bluffs based on previous research and observations.
Thus, probability and statistics play a vital role in poker. The previous example I have illustrated is very basic, and there are many more additional variables that go into calculations. For example, if there are many players, the conditional probability would have to take all actors into account. In addition, the probabilities depend on the type of poker game being played and the round they are in (e.g. the probability during the river of Texas Hold ‘Em would be different than the probability during a 7-card draw poker game). Furthermore, we could think about the maximum amount we can bet in order to gain the largest expected payoff. Lastly, as we have learned in class, we could think about how information cascades could affect each player’s decision. For example, if player 1 bets, player 2 might fold based on his hand. Player 3 might have a really good hand, but he might make the wrong decision because of player 2’s actions. Every great poker player has to make quick changes to their complicated calculations in order to succeed, and it is truly impressive.