Expected Value and Fair Games Reference Cheat Sheet

Key Facts

• The expected value of a discrete random variable is $E(X)=\sum x_iP(x_i)$.
• To find expected value, multiply each outcome by its probability, then add the products: $E(X)=x_1P(x_1)+x_2P(x_2)+\cdots+x_nP(x_n)$.
• A probability distribution is valid only if every probability satisfies $0\le P(x_i)\le 1$ and the total probability is $\sum P(x_i)=1$.
• A game is fair when the expected value for a player is $E(X)=0$.
• If $E(X)>0$, the game has a long-run expected gain for the player receiving the payoff.
• If $E(X)<0$, the game has a long-run expected loss for the player receiving the payoff.
• Net payoff equals winnings minus cost, so $\text{net payoff}=\text{prize}-\text{cost to play}$.
• Expected value describes the average outcome over many trials, not the guaranteed result of one trial.

Vocabulary

Expected Value

The long-run average value of a random variable, found using $E(X)=\sum x_iP(x_i)$.

Random Variable

A variable whose value depends on the outcome of a chance process.

Probability Distribution

A table or rule that lists all possible values of a random variable and their probabilities.

Payoff

The amount gained or lost from an outcome, often calculated as net winnings after subtracting the cost to play.

Fair Game

A game with expected value $0$, so no player has a long-run advantage.

Long-Run Average

The average result expected after a large number of repeated trials.