To many people poker is just a game of cards. The mass-media typically depicts poker as a game of big bluffs and crazy luck. Some high profile successful poker players say they don’t know much about the math or probabilities and play by instincts and experience. A lot of regular players can’t be bothered with pot-odds and just play their cards. What many of these people don’t realize is that everything in poker is governed by mathematics. <READMORE>

I’m not going to go into too much mathematical detail in this article. Bill Chen and Jerrod Ankenman have written a book, The Mathematics of Poker, where there are lots of mathematical details. Instead, I’m going to try to explain in layman’s terms why poker is governed by mathematics and why, even if you didn’t realize it, every poker action you take has a mathematical basis.

The first thing to realize is that poker is not so much a game of cards as a series of bets. Without bets, any form of poker would just be a random game of luck where the best hand wins. With bets, however, you have the ability to influence the outcome of a hand by choosing whether to continue to participate in a hand or not.

When you choose to bet or call, what you are really doing is making a wager from which you hope to profit. An easy way to understand this is calling an all-in bet on the river holding the stone-cold nuts. Your opponent has effectively offered you a wager for all of his chips plus those already in the middle. No matter how much he has bet it is always profitable to accept this wager. You have a 100% probability of winning the wager and the odds you are offered on the wager can never be worse than even money. You only need to win an even-money wager 50% of the time to breakeven so by the mathematics 100% > 50% and it is correct to call.

I know that some of you are thinking that example is so obvious and there’s no need for the mathematical proof, but now consider this case. A once-in-a-lifetime promotional contest for a million dollars holding AA on a KKAQ board with $980,000 in the pot and your last $20,000 to call. Your opponent shows you KK and says, “I’ll give you $20,000 if you fold and concede the contest.” Many people would instinctively accept this deal, figuring they would rather have $20,000 than take the long-shot that the case Ace hits the river. Mathematically, however, the wager being offered here is betting $20,000 to win $1,000,000. You only need to win this wager 2% of the time to break-even. There are 52 cards in the deck, 8 have already been exposed and 1 of the remaining 44 cards is an Ace. There is a 2.3% chance that the Ace hits the river. Since 2.3% > 2.0% it is profitable to decline the deal and call the bet.

Pretty much every poker action you take will fall between these two mathematical extremes and is governed by the mathematics. When you call a bet on the river with a good hand, you’re choosing to take the odds being offered by your opponent to win the pot. You might just decide to guess that your hand is good enough or simply take a chance with lady-luck, but if you want to be a consistently profitable player, you should give some consideration to whether your hand will win the showdown a greater percentage of the time than necessary to profit from the wager you’ve been offered.

Similarly, when you bet on the river and your opponent calls, you are the one offering the wager to your opponent, and you should give some consideration to how the size of your bet will affect your opponent's ability to call profitably. Your aim here is to bet an amount where he wins a lower percentage of the time than his breakeven rate for accepting the wager.

Bets and calls made before the river act as a precursor to a showdown and establish the subsequent odds. The mathematics are much more difficult when you need to factor in the probabilities of different combinations of potential cards coming on subsequent streets, but fundamentally, each bet or call should have a mathematical basis for being profitable or unprofitable.

"But what about pots won without showdown? Surely there is more to those pots than the math," I hear you cry. Well, these pots fall into two categories: betting for value and betting for deception.

When you bet for value and your opponent folds, you may have won the pot, but you potentially lost some profits. You should always try to bet an amount where the wager being offered to your opponent is unprofitable to them (given the information that you know but that they may only suspect), but not so much that this mathematical fact becomes obvious to them.

Take, for example, flopping top set on a Kd 7d 2s board. You know you have the best possible hand right now, but there are many hands that your opponent might be holding that could beat you on the river. He could be holding a flush draw with the Ad, and while you know he only has a 25% chance of winning at showdown, he may think he has a 40%+ chance. To make a mathematically correct bet, you should be offering him no higher than 3-to-1 odds and no lower than 3-to-2 odds. Any bet you make outside of this range is either profitable for him to call or easy for him to fold (this applies if he is a smart, thinking player; you may be able to get bigger bets called by weak players). When value-betting, therefore, you are potentially losing out on profits if you don’t take into account the mathematics of your action and bet accordingly.

Finally, when you are betting for deception (bluffing), isn’t this just about fearless aggression, psychology, and picking on weak opponents? Well, yes it is, but there’s still a fundamental mathematical basis to it. When you bluff, you are trying to win a pot that you would otherwise expect to lose. You need your bluffs to be successful a sufficient percentage of the time to be profitable.

Take the easiest example of bluffing on the river with a hand that otherwise cannot win a showdown. If you bet the pot, this bluff needs to be successful more than 50% of the time to be profitable. Once again, you are simply making an even-money wager that will be mathematically correct if you deduce that your opponent will fold more than 50% of the time. All other bluffs on other streets have a similar mathematical basis, including a 4-bet all-in pre-flop shove; it's just that there are the added complexities of the cards to come potentially winning you the hand even if called (i.e. it is semi-bluffing when there is a chance you might win at showdown, and this factor is part of the mathematical basis for such bluffs).

All told, therefore, the mathematics of poker are omnipotent. There is no situation or action where mathematic laws and probabilities do not play a major role in the outcome. Yes, there are psychology and intuition to help you determine how often certain plays will succeed, but there are always underlying mathematics to back them up, even if you didn’t realize it.
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