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Significance of Expected Value

Expected value (EV) is a central principle in the theory of probability. It is used for average estimation of some random value. Expected value is similar to a center of gravity assuming that the values of probability are the masses of solid point.

Let us assume that the game has n different outcomes, probability of each outcome is p_i. The expected value of some variable x that takes values x_i can be calculated with the next formula:

E(x) = x₁p₁ + x₂p₂ + ... + x_np_n ,

In case the probabilities of outcomes are equal ( p₁=p₂=...=p_n=1/n ) the expected value equals to arithmetic mean:

E(x) = x₁/n+ x₂/n + ... + x_n/n = (x₁ + x₂ + ... + x_n) / n

Why is the expected value the most important principle in the probability theory? It helps to predict the estimation of some random variable for a long period of trials. The mean of any random variable in a long term gets close to its expected value. This fact is strictly proved in the course of probability theory.