Expected value is the average result you would expect from a random process if it were repeated many times. It does not have to be an outcome that can actually occur in one trial. Instead, it summarizes the balance of all possible outcomes and their probabilities.
This idea matters in games, insurance, finance, science experiments, and any situation where decisions involve uncertainty.
To find expected value, multiply each outcome by its probability and add the products. Outcomes with larger probabilities have more influence on the final average, like heavier blocks on a balance scale. A fair game usually has expected value 0 for a player, while a favorable game has positive expected value.
In real applications, expected value helps compare choices, but it does not describe the risk or spread of possible results by itself.
Key Facts
- For a discrete random variable X, E(X) = sum of xP(x) over all outcomes.
- Expected value is a weighted average, where probabilities are the weights.
- All probabilities in a probability distribution must satisfy 0 <= P(x) <= 1 and sum P(x) = 1.
- Expected value can be negative, zero, or positive depending on the values of the outcomes.
- Linearity of expectation: E(aX + b) = aE(X) + b.
- Net expected value in a game = expected winnings minus cost to play.
Vocabulary
- Random variable
- A random variable is a quantity whose value depends on the outcome of a random process.
- Expected value
- Expected value is the long-run average value of a random variable over many repeated trials.
- Probability distribution
- A probability distribution lists each possible value of a random variable and the probability of that value.
- Weighted average
- A weighted average combines values so that values with larger weights have a greater effect on the final result.
- Fair game
- A fair game is a game whose net expected value is 0 for the player.
Common Mistakes to Avoid
- Averaging the outcomes without probabilities: This is wrong because expected value gives more influence to outcomes that are more likely.
- Forgetting to subtract the cost to play: This gives expected winnings, not net expected value, so it can make a losing game look profitable.
- Using probabilities that do not add to 1: A valid probability distribution must include all possible outcomes with total probability 1.
- Interpreting expected value as a guaranteed single-trial result: Expected value describes a long-run average, not what must happen on the next trial.
Practice Questions
- 1 A spinner pays 5 with probability 0.30, and $10 with probability 0.20. What is the expected payout?
- 2 A game costs 12 with probability 0.25 and win $0 with probability 0.75. What is the net expected value for the player?
- 3 Two games have the same expected value, but one has a small chance of a very large prize while the other gives a steady small prize often. Explain why a player might prefer one game over the other even though the expected values are equal.