Note for pokertips.org forum readers - i’m just hosting this here temporarily for someone to read, i’m aware you all know what expected value is.

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Expected Value is a term used to describe a player’s expected return on a wager in a pot. Players should attempt to make plays with a positive expected value (+EV) and avoid situations with a negative expected value (-EV) for the player.

For Example

All players have 5,000 chips with blinds at 250/500.

Action is folded to Player A on the small blind. Player A raises all-in.

Player B is on the big blind. It is 4,750 chips to call to win a 10,000 chip pot, meaning that Player B needs to have 47.5% equity against Player A’s likely shoving range in order to yield a positive expected value by calling. This is calculated by dividing the amount needed to call into the total pot size.

Player B estimates that Player A is moving all-in with a range of 66-AA, AT-AK, KQ.

Player B has Qs Qc.

If Player A has a pocket pair, there are six situations where he will be an 80% favourite, one unlikely situation where they have the same hand (which is so unlikely it barely figures into calculations since Player B has two of the queens) and two situations where he will be a 20% underdog. He can estimate his equity here by multiplying 80% (his approximate percentage to win when he is ahead) by 6 (the number of times he is ahead on average when Player A has a pocket pair) and 20% (his approximate percentage to win when he is behind) by 2 (the number of times he is behind on average when Player A has a pocket pair) and adding the two numbers together to get 520, then dividing by the number of hands (eight hands) to get an average of 65% equity when Player A has a pocket pair.

If Player A doesn’t have a pocket pair, there is one situation where Player B is flipping (against AK) and four situations where Player B is a 70% favourite. Using the same formula, Player A multiplies 50% by 1 and 70% by 4, adds them together to get 330 and then divides by 5 to get 66% equity when Player A does not have a pocket pair.

Because there are more combinations of non-paired hands than paired hands, even though there are more hands in the pocket pair part of Player A’s range, Player B estimates that Player A will have a pocket pair roughly half the time and high cards the other half of the time. Combining his estimates so far, Player B averages out his equity when Player A does and does not have a pocket pair to estimate that he has 65.5% equity against Player A’s shoving range.

Because Player B’s estimated equity of 65.5% is greater than the required 47.5% to make the call, calling here is a play with a positive expected value because he has significantly more than 47.5% equity against Player A’s range. In fact, if the raise was not all-in, Player B has so much more equity with his hand than Player A does that he could even reraise. However, if Player B had a hand like 6c 5c, his equity would be far lower than 47.5% so calling would have a negative expected value in this instance.

Of course, these calculations are not precise, but using this method it becomes easy for players who are capable of estimating their opponents ranges to determine whether calling is likely to yield a positive expected value in the long run in situations where your opponent has bet or raised with no further action pending and you are considering a call.

If there is further action pending on the hand, additional calculations must be factored in, including implied odds and reverse implied odds before a player can calculate whether calling, raising or folding is the correct course of action. For further details, see implied odds.