Expected value helps students predict the long-run average result of a random process, game, or decision. This cheat sheet covers how to calculate expected value from outcomes and probabilities, how to read payoff tables, and how to decide whether a game is fair. Students need these tools for probability, statistics, finance, and real-world decisions involving risk.
The main formula is , where each outcome is multiplied by its probability and the products are added. A fair game has expected value for each player, meaning no player has a long-run advantage. Positive expected value favors the player receiving the payoff, while negative expected value means an expected loss over many trials.
Key Facts
- The expected value of a discrete random variable is .
- To find expected value, multiply each outcome by its probability, then add the products: .
- A probability distribution is valid only if every probability satisfies and the total probability is .
- A game is fair when the expected value for a player is .
- If , the game has a long-run expected gain for the player receiving the payoff.
- If , the game has a long-run expected loss for the player receiving the payoff.
- Net payoff equals winnings minus cost, so .
- 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 .
- 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 , so no player has a long-run advantage.
- Long-Run Average
- The average result expected after a large number of repeated trials.
Common Mistakes to Avoid
- Using prize amounts instead of net payoffs is wrong because the cost to play must be subtracted before calculating expected value.
- Forgetting to multiply each outcome by its probability is wrong because expected value is a weighted average, not a simple average of outcomes.
- Using probabilities that do not add to is wrong because a probability distribution must include all possible outcomes exactly once.
- Thinking a positive expected value guarantees a win on one play is wrong because expected value describes long-run behavior, not a single trial.
- Calling a game fair when the prizes look equal is wrong because fairness depends on , not on whether the outcomes seem balanced.
Practice Questions
- 1 A game costs \2\ with probability and win \0\frac{4}{5}$. What is the expected net payoff?
- 2 A spinner has outcomes , , and with probabilities , , and . Find .
- 3 A raffle ticket costs \5\frac{1}{100}\ and a chance of winning nothing. Is the raffle fair based on expected net payoff?
- 4 Why can a game with still allow a player to win money on a single turn?
Understanding Expected Value and Fair Games Reference
A payoff table must describe the money that actually changes hands. If a ticket costs two dollars and a player wins five dollars, the payoff is a gain of three dollars, not five dollars. This detail is the source of many wrong answers.
Include the ticket cost in every possible result, even when the player loses. It helps to make separate columns for the outcome, its probability, the prize, and the net payoff.
Then check that every possible outcome appears once. For a spinner or card draw, outcomes that give the same payoff can be combined by adding their probabilities.
Expected value comes from weighting outcomes by how often they occur. A rare large prize may look important, yet its contribution can be small if its probability is tiny. A common small loss can have a much stronger effect because it happens repeatedly.
For example, a game might pay twenty dollars with probability one tenth and lose three dollars with probability nine tenths. The large prize contributes two dollars to the average. The losses contribute negative two dollars and seventy cents.
The game therefore gives a negative average result of seventy cents per play. This explains why many games can offer exciting prizes while still earning money for the organizer.
Fairness depends on the viewpoint. If a game is fair for the player, the player has no average gain or loss after the entry cost is included. The organizer then has no average gain or loss either, if there are no extra costs.
In real businesses, organizers need money for staff, equipment, and profit. Their games usually have a negative expected value for customers. Lotteries, carnival games, prize machines, and many online game purchases work this way.
Insurance is different but uses the same reasoning. A customer accepts a small certain cost to protect against a rare, expensive loss. The insurance company uses expected values across many customers to set premiums.
A long-run average does not make short runs predictable. A player can win several times in a row in a game with a negative expected value. Another player can lose in a game that is fair.
Random variation is especially large when there are few trials or when prizes vary widely. As the number of trials grows, the average result usually moves closer to the expected value. Students should not treat this as a promise that losses will soon be balanced by wins.
Past results do not change the probability of the next independent roll, spin, or draw. When checking work, verify that probabilities total one, use net payoffs rather than prizes, keep negative signs for losses, and state whose gain or loss the answer represents.