Poker Strategy - Expected Value

If you comb through poker strategy books or articles written on this website, you will repeatedly find assertions that poor poker players may win money in the short run, but lose money in the long run. The converse is true for professional and very good players, who will generally win in the long run.

Why is this exactly? It is because of a concept known as expected value. Expected value is your expected return on a wager. For example, suppose you made a bet with me on a coin flip. If it is heads, I give you $100. If it is tails, you give me $1. Should you theoretically take this bet (assuming that the coin is fair and has a fifty-fifty chance of landing on heads or tails)?

Obviously, you should make the bet. There is a 50% chance of it landing on heads, meaning you win $100. Thus, your expected win is $50 (0.50 * $100). If it lands on tails, you lose $1. Thus, your expected loss is $0.50 (0.50 * $1). Your expected profit is the expected win minus the expected loss. Thus, your expected profit is $49.50.

Obviously, you will not win $49.50. You will win $100 or lose $1. However, you should view the bet as "winning" $49.50. Outcomes in gambling are influenced by chance in the short run. However, in the long run, your outcomes will very closely reflect your expected value. If we did the coin flip example a million times, your final profit would be extremely close to $49.5 million.

So, how does expected value play out in a poker game? The most obvious manifestation of expected value is pot odds theory (pot odds, as well as implied odds, reverse implied odds, effective odds, etc.). The whole idea of pot odds theory is that you should only draw to a hand when you have the best of it (when you have a positive expected value when you draw.)

Other examples of expected value are demonstrated in some advanced moves.


Your Hand		Board

There are 8 players in the pot and you are the small blind (first person to bet). You check, and the player in the big blind position bets. Three players call. Should you call?

No, you should raise! Why? There is a 35% chance you will hit a flush on the turn or the river. If you hit this flush, you will more than likely scoop the pot. If you raise, 4 players will call, meaning you will only be putting in an additional 20% of the pot. Thus, if this was a $1-$2 game, you would have an expected value profit of $.75 (0.35 * $5 - $1).

Please realize that this is an imperfect example. The expected win only considers the flush draw. You also have overcards, which means you could easily win if a ten or an Ace falls as well. However, there is also the chance that someone may hit a full house, so you'd have to decrease it by that. Someone could also reraise behind you and knock others out, which may decrease your expected value. Nonetheless, hopefully this example shows why some fancy plays are made.

Bluffing and calling bluffs depend on expected value. When you make a bluff, you should have some estimate in your head about the bluff's chance of success. This chance of success should meet with a positive expectation. For example, if the pot is $100 and I make a bluff of $50, I need at least a 33% of winning (assuming I have 0% chance of winning if I am called). This is because I need to win 1 out of 3 times in order to break even.

Expected value also helps clarify the differences between big mistakes and small mistakes. Big mistakes occur when a player makes a decision that has a very high negative expected value whereas a small mistake is when one gives up a small amount of expected value. See When to Fold for more detail about situations where players make small or big mistakes.

If you plan on playing poker a lot, you should eventually be able to know your expected value of an hour of play. In order to do this, you need to keep accurate records. You should record the amount of time you spend playing, the type of game, and the place where you played. You can do this using specialized poker recording software, such as Poker Charts. After a while, you will be able to reasonably estimate your expected win or loss per hour of play (you need to log at least 200 hours at a given location/limit to have a decent estimate). You can refer to the keeping records article.

Expected value is another reason why you should never play in a poker game that you cannot afford. If you play with scared money, you will be reluctant to play when you have a small edge. You will give up a lot of expected value on some hands, which will probably turn you into an overall loser.

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