Expected Value

[BB] “Every action you take in a poker game has an expected value, which is the average amount of chips or money that action will win or lose in the long run”, Benny the Book explained to his son.

EV of Folding

[BB] “Folding has an expected value, or EV, of 0, since you aren’t gaining or losing any more chips. Remember that the chips in the pot are no longer yours, so they don’t factor in when you fold. Whether there are 10 chips or 10,000 chips in the pot, folding will change your chip stack by the same amount, zero.”

EV of Calling

[BB] “Following up on our discussion of pot odds, the EV of calling is calculated as the odds of winning times the pot size minus the odds of losing times the call size. For example, suppose the pot is P and you call a half-pot all-in bet on the turn when you’re sure you need to hit a flush on the river. Since 9 outs is about 20%1, the EV of the call is 0.2 * 1.5P – 0.8 * 0.5P = -0.1P2. You’ll lose just over a tenth of the original pot size on average by making this bad call. If you also had a straight draw, you have 15 outs, about 32.5%, which we can call 1/3, and (1/3)*1.5P – (2/3)*0.5P = 1/6P3. You’ll win almost a sixth of a pot on average.”

EV of Raising

[BB] “This is impossible to calculate without knowing how often your opponent is going to fold, call, or reraise, but we can lay out the general formula. For simplicity, let’s say we’re on the river, the pot is P, your opponent bets B, you raise R, and you have the better hand N% of the time (all percentages should be divided by 100 to give a number between 0.0 and 1.0). When your opponent folds, you win P+B. When your opponent calls you win P+B+R if your hand is better and lose B+R if your hand is worse. Let’s ignore the possibility of your opponent reraising for now. Your overall EV is (opp fold %) * (P+B) + (opp call %) * ((N * (P+B+R)) – ((1-N) * (B+R))).”

[JJ] “You expect me to crunch all that at the table?”, Joey the Juvenile objected.

[BB] “No, but it’s useful in postgame analysis. The important thing to note is that your raise wins some of the time by folding your opponent and some of the time when you have the best hand. Whenever you raise, you want to think about both of those possibilities to determine how much to bet. Against calling stations, you should expect fewer folds, so you should bet more for value. Against tight players who haven’t shown any strength, you can bluff more.”

[BB] Seeing his son smiling much too broadly, Benny regretfully wondered, “Why do I feel like I’m training a maniac?”

Footnotes:

  1. This was covered a few days ago in Counting Outs.
  2. The exact calculation is 9/46 * 1.5P – 37/46 * 0.5P equals about 0.29P – 0.40P = -0.11P.
  3. The exact calculation is 15/46 * 1.5P – 31/46 * 0.5P equals about 0.49 – 0.34 = +0.15P.

Related Links:

  • Bobby Crosby’s PG-13 +EV comic follows the life of an addicted online poker player, who views everything in life in terms of Expected Value. Sometimes funny, sometimes sad. Always depraved.
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