Poker/Expected value

Expected Value (EV) is the average return on each dollar invested into a pot. If a player can expect, given probability to make more money than he or she bets, the action is said to have a positive expectation (+EV). Conversely if a bet or a call will, according to probability, likely result in less money being returned the action is said to be negative (-EV).

An example may assist in the understanding of this concept. In Texas Holdem it is quite common for someone to flop 4 to a flush. The person should only draw to that flush if to do so would be +EV. In order to calculate the EV it is necessary to compare the size of the bet with the size of the pot. A flopped flush draw will come in approximately 1 in 3 times by the river, thus in order for a call to be +EV the final pot must be larger than 3 times the call. This is a complicated issue so it may be useful to elaborate with a specific example.

Say you are playing 5-10 limit poker on the button, there are 3 limpers to you and you call with A4 diamonds. Both Blinds call so there is $30 in the pot. You flop the nut flush draw. The player in the small Blind bets $5 and there are four callers. Should you call, raise or fold?

Well, there is now $55 in the pot and it will cost you $5 to call so the pot is giving you 11:1 odds (i.e. you must pay $5 to win $55). We already know that the flush draw will get there 1 time in every 3 (2 to 1) so making the call is +EV. However, calling is not necessarily the best play in this situation. If you raise and the other 5 people in the pot decide to call your raise then you will be adding $5 to the pot whilst they will collectively be adding $25. This ratio is 5:1 but the chance of making your flush is only 2 losses to 1 win, so on average in the long run you are making money from every extra bet from all 6 of you that goes into the pot. Notice that even though you have only a 33% chance to win the pot, the correct thing to do is actually to bet, despite knowing that you will probably not win that particular single hand: you will win about 1 in 3 such hands in the long run, minus the few percent of the time when someone beats your flush with a full house or quads or straight flush.

Notice that all that has been done so far is compare the current pot with the bet size needed to call to calculate EV. However it is important to also compare the expected pot size by the end of the hand with the current bet. For example say you are playing no limit holdem and have a gutshot straight draw (giving you 4 outs to complete - approximately 1:12 against). If the pot is $30 and you are faced with a $10 bet the pot is not giving you the correct odds to call (it would need to be $120 total, plus your $10 call). However you also need to take into account the amount of money you may be able to extract from your opponents if you make your hand. If you expect your opponent to call a $100 bet if you make your hand, then the pot is really offering you 13:1 odds (the $30 pot at the time plus the $100 added on later streets) Therefore in this situation the +EV play would be to call. Thus when making decisions about whether to call a bet it is crucial to take into account both the stack sizes of yourself and your opponents and how willing they are likely to be to call big bets if you make your hand.

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. Casinos worldwide make MILLIONS of dollars lost by players against a 0.6% craps dice game house edge: even 0.1% is enough of an edge to wipe out all the billions of dollars of the richest man on earth, over time in the long run, which is why Bill Gates wisely bets only $5/hand for fun at blackjack!

Calculating expected value

You cannot always get a good idea of the chances of winning your hand & calculate the pot odds: 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!