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Calculate expected value

Expected value is the probability-weighted mean of the possible outcomes of a discrete random variable, calculated from a probability table or distribution using E(X)=∑xP(X=x)E(X)=\sum xP(X=x), where the probabilities sum to 1. It represents the long-run average outcome over many repetitions, not the outcome most likely on a single trial or a guaranteed result, and supports analysis of games, decisions, and fairness. Continuous-distribution integrals and more advanced expectation theory are beyond this scope.

Detailed Explanation: Calculate expected value

The expected value is the probability-weighted mean of all possible outcomes. It tells you the average result you would expect over many repetitions.

Use the formula

E(X)=∑xP(X=x)E(X)=\sum xP(X=x)

This means:

  1. Multiply each outcome xx by its probability.
  2. Add all the products.
  3. Check that the probabilities add to 11.

Worked example

A game has the following possible winnings:

Winnings xxProbability P(X=x)P(X=x)
$00.50.5
$50.30.3
$200.20.2

First, check the probabilities:

0.5+0.3+0.2=10.5+0.3+0.2=1

Now multiply each outcome by its probability:

E(X)=(0)(0.5)+(5)(0.3)+(20)(0.2)=0+1.5+4=5.5\begin{aligned} E(X) &=(0)(0.5)+(5)(0.3)+(20)(0.2)\\ &=0+1.5+4\\ &=5.5 \end{aligned}

Therefore, the expected value is

$5.50\boxed{\$5.50}

This does not mean you will win exactly 5.50inonegame.Itmeansthatifthegamewereplayedmanytimes,theaveragewinningspergamewouldapproachin one game. It means that if the game were played many times, the average winnings per game would approach$5.50$.

Learn by doing: Calculate expected value

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    Probability Random Variables - Probability Table to Expected Value Calculation

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