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Expected Value (EV) - Guide explaining the concept of expected value.

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Expected Value / EV

Expected value can get a little complex however the basics are quite straight forward. In principle every poker player will be faced with decision after decision in how they play each hand. Expected value is the return you can expect to receive over the long run based upon the probabilities of various outcomes. EV is affected not only by you but also the players around you and can be positive (meaning you make money in the long run) or negative (you lose in the long run). So why is this so important?

Well every player should want to maximize their expected value as they should be looking to maximize their return on investment (how much they wager). Check out this article on Wikipedia which goes into significant depth on expected value.

"In probability theory and statistics, the expected value (or expectation value, or mathematical expectation, or mean, or first moment) of a random variable is the integral of the random variable with respect to its probability measure. For discrete random variables this is equivalent to the probability-weighted sum of the possible values, and for continuous random variables with a density function it is the probability density -weighted integral of the possible values.

The term "expected value" can be misleading. It must not be confused with the "most probable value." The expected value is in general not a typical value that the random variable can take on. It is often helpful to interpret the expected value of a random variable as the long-run average value of the variable over many independent repetitions of an experiment.

The expected value may be intuitively understood by the law of large numbers: The expected value, when it exists, is almost surely the limit of the sample mean as sample size grows to infinity. The value may not be expected in the general sense – the "expected value" itself may be unlikely or even impossible (such as having 2.5 children), just like the sample mean. The expected value does not exist for all distributions, such as the Cauchy distribution.

It is possible to construct an expected value equal to the probability of an event by taking the expectation of an indicator function that is one if the event has occurred and zero otherwise. This relationship can be used to translate properties of expected values into properties of probabilities, e.g. using the law of large numbers to justify estimating probabilities by frequencies."

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