Poker:Expected value

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The expected value (EV) or expectation of a wager is how much you will win in the "long run" by making that wager. For example, this situation comes up often in Texas hold'em: a player has four hearts on the turn, and needs to hit a heart on the next and final card to make a flush. He is absolutely certain he will win if he hits the flush and in so doing the board does not pair, because the board allows no possibility for a full house or better hand, and his flush will be the nut flush. Even if the board pairs, he has a good chance of winning, so he need not worry about other hands that much. The probability of catching the heart is about one in five. Therefore, he can expect to lose most of the time. However, the idea of continuing cannot be dismissed out of hand unless we know what his expectation is. If it will cost $5 to call a bet and the pot, including other bets and calls, is $50, our player actually has a very positive expectation and should pay to see the next card. The chance of hitting his hand is only one in five, the pot is ten times the bet size. The player expects to win on average two dollars for every dollar invested every time this situation occurs.

To make it easier to understand why this move is correct even though it usually loses, suppose you have a six-sided die. If you correctly guess what side it lands on, you will win $50. If you are wrong, you lose $5. You will be wrong five times out of six, but you stand to gain a lot over the long run! This is because the probability of guessing correctly is 1/6, sometimes expressed as odds, "5:1 against" (five losing possibilities, one winning possibility). However, the payoff odds are 50:5 ($50 won for a $5 bet), which can be reduced to 10:1, and 10:1 is twice as large as 5:1. The payoff odds are called pot odds in a poker game. Comparing the odds of winning to the pot odds is how you can estimate your expected value.

Ideally, you want to avoid all situations where you have a negative expectation. Even slightly negative situations can pile up and bleed away your bankroll.

[edit] Calculating expected value

You cannot always get a good idea of the chances of winning your hand — at least, not without knowing what your opponents have, and they're not going to tell you! However, you will often have a draw which, if you hit, you will very likely win the pot. The exact arithmetic involved varies from game to game. In Texas hold'em and Omaha, once you see the flop, the percent chance of making your hand within one card is generally your number of outs (cards that will make your hand) multiplied by two, and the odds of making your hand within two cards is your number of outs multiplied by four. For example, if you have four hearts and you need one more for a flush, you have nine outs, because there are thirteen hearts in the deck, and subtracting the four hearts you already have gives nine. 9 × 2 is 18, so you have about an 18% chance of making the hand in the next card, and 9 × 4 is 36, so you have about a 36% chance of making it in two cards.

To make this easy, you want to turn this percent chance into odds, like 5:1 against. Fortunately, they are easy enough to memorize:

50% =  1:1
33% =  2:1
25% =  3:1
20% =  4:1
16% =  5:1
14% =  6:1
12% =  7:1
11% =  8:1
10% =  9:1
 9% = 10:1
 8% = 11:1
 7% = 13:1
 5% = 20:1
 4% = 25:1

The odds in bold are the most important to commit to memory; the others can be easily estimated.

Now, take the x in the x:1 figure and multiply it by the bet size. For example, if the odds of making your hand are roughly 4:1, and the next bet costs $5, multiply 5 × 4 = 20. That means you want there to be at least $20 in the pot (be sure to include bets that have not been added to the pot proper yet!), preferably a bit more just in case unless you're certain to win if you hit your draw. If there is not at least $20 in the pot you will lay down your hand, unless you can check instead. If the table is really loose, and a lot of players are in the hand and are likely to stay in, and the pot will get really big, you may even want to raise. Normally, however, checking or calling is the correct move.

Notice we did not calculate the exact expected value. This is not necessary or indeed practical for most people. If it is negative, you get out, and if it is positive, you call. If you're a favorite to win the pot, you raise. However, as has been shown you can usually figure out if the value is only barely positive, for instance, the size of the pot is a dollar more than the odds of making your hand (and this dollar is small in proportion to the pot size). When faced with this situation, you might want to lay down your hand sometimes: you may be losing just a little money in the long run, but you keep your bankroll from taking big swings. But if you don't mind taking a gamble, by all means go for it!