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Expected Value In Poker

This instablog was written for the January 2013 Wildcat Investors Newsletter.

"Investing is a probabilities game, not a certainties game" - Ken Fisher

Investing and poker share many attributes. One of the decision making tools I utilize in both disciplines is expected value. Expected value provides the ability to make decisions when faced with probabilities. In practice, using expected value in a poker hand, as well as investing, rewards those that are able allocate accurate probability weights.

I will examine the art and science of expected value through a poker hand I played last April. There were three limpers before me. I limped from late position with K 6 off suit. The cutoff (person to my left) called and blinds completed. The pot is $14. Flop: 7s 6c 2c. It limped around to me and I bet $11. The cutoff called and everyone else folded. The pot is $36. Turn: j hearts. I bet $22 and he raised to $75. I called. The pot is $186. River: 2. The board was 7 6 2 j 2 with no flush possibility. I checked and my opponent goes all in for $125. The pot is $311.

There are two primary variables to calculating expected value; the amounts won/lost, and the probability of winning/losing. The amount won/lost is fairly straightforward to calculate. If I win, the payoff is $311 to a $125 investment; equating my breakeven to (2.5/1), or approximately 30%. The difficult part is assigning accurate probabilities given the information. Information can come in many ways, betting patterns, past history, tells, etc. I like to take each piece of information as it moved through time separately, and then cross reference it. That means looking at how he played the hand preflop-flop-turn-river individually, than as a complete puzzle.

The way he played, I am confident he has value combinations (hands that beat me) of 67, 6j, 7j, or 22. Bluff combos are 89,45,58, 45c,34c,9tc,tjc,jqc,q9c,q8c,a3c,a4c,a5c. As you can see, the bluff hands outnumber the value hands. In fact, I have approximately 65% equity with just my pair of sixes against his estimated range (Remember, this is a blend of subjective & a priori probability).

To calculate expected value from the call, I multiply the payoff with the probability of success, and then subtract the product of the amount loss with the probability of loss. This looks like: (311*.65 - 125*.35) = $158. This means that every time I make this call, I theoretically make $158, regardless of what hand he shows.