Fair Game — Definition, Formula & Examples

A fair game is a game where every player has an equal chance of winning or, more precisely, where the expected value of each player's net gain is zero. No one has a built-in mathematical advantage over anyone else.

A game is called fair if the expected value of a player's net winnings equals zero. That is, when you multiply each possible outcome's payoff by its probability and sum the results, the total is $0$ .

Key Formula

E = p_1 \cdot v_1 + p_2 \cdot v_2 + \cdots + p_n \cdot v_n = 0

Where:

$E$ = Expected value of net winnings (must equal 0 for a fair game)
$p_i$ = Probability of outcome i
$v_i$ = Net gain or loss for outcome i

How It Works

To check whether a game is fair, calculate the expected value of one player's net gain. List every possible outcome, note its probability and its payoff (positive for a win, negative for a loss). Multiply each payoff by its probability, then add them up. If the sum equals zero, the game is fair. If the expected value is positive, the game favors that player; if negative, it favors the opponent.

Worked Example

Problem: You and a friend flip a coin. If it lands heads, your friend pays you $3. If it lands tails, you pay your friend $3. Is this a fair game?

List outcomes: Heads: you gain $3. Tails: you lose $3. Each has probability 1/2.

P(H) = \frac{1}{2}, \quad P(T) = \frac{1}{2}

Calculate expected value: Multiply each payoff by its probability and add the results.

E = \frac{1}{2}(3) + \frac{1}{2}(-3) = 1.5 - 1.5 = 0

Answer: The expected value is $0, so this is a fair game — neither player has a mathematical advantage.

Why It Matters

Understanding fair games helps you analyze whether real-world games, bets, or lotteries are stacked against you. Casino games and state lotteries, for example, are not fair — their expected values are negative for the player. This concept appears frequently in middle school and high school probability courses.

Common Mistakes

Mistake: Assuming equal probability of winning automatically means a game is fair.

Correction: A game is only fair if the expected value of net winnings is zero. Two players could each have a 50% chance of winning, but if one wins $10 and the other only wins $5, the game is not fair.