Math101 · Independent Ontario learning library
Math101learn.math101.caExpected Value
Expected value is the probability-weighted long-run average of a random variable.
Expected value balances every possible outcome by how often the model says it should occur in the long run.
Definition
For a discrete random variable $X$ with possible values $x_i$,
Multiply each value by its probability, then add. The probabilities must represent a complete valid distribution.
Long-run interpretation
If the random process is repeated independently many times under stable conditions, the average outcome tends to approach $E[X]$.
Expected value need not be a possible single outcome. The expected number on a fair die is $3.5$, even though no roll shows $3.5$.
Worked example: a simple game
This does not mean each play earns ten cents; it describes the long-run average.
Fair games
A game is mathematically fair to the player when the expected net gain is zero. Positive expectation favours the player; negative expectation favours the organizer.
Include entry costs and all payouts consistently. A prize amount is not the same as net gain if the player paid to enter.
Expected value from counts
If outcomes are equally likely, probabilities can be obtained from counts. For a fair six-sided die,
For unequal probabilities, a simple arithmetic average of possible values is incorrect; use the weights.
Linearity of expectation
For constants $a,b$,
More generally,
even when $X$ and $Y$ are not independent. Independence is needed for some variance rules, not for this sum rule.
Binomial expectation
If $X\sim\operatorname{Bin}(n,p)$ counts successes in $n$ independent trials, then
This makes sense: each trial contributes expected success count $p$, and the expected counts add over $n$ trials.
Decision-making and risk
Expected value compares average financial or quantitative outcomes, but it does not capture spread, worst-case consequences, utility, or ability to tolerate loss.
Two choices can have the same mean and very different variance. High-stakes decisions require more than expectation alone.
Empirical expected value
Observed data can estimate expectation using a weighted average of values by relative frequency. The estimate may vary from sample to sample and may be biased if data collection is unrepresentative.
Distinguish the theoretical model mean from a sample mean used to estimate it.
Insurance interpretation
Insurance pools many uncertain losses. An expected claim cost helps set premiums, but administrative costs, capital requirements, risk, and profit also matter.
An individual may rationally pay more than expected loss to reduce exposure to a rare severe outcome.
Common mistakes
Averaging possible values without probability weights. Outcomes may not be equally likely.
Using prize instead of net gain. Subtract any cost consistently.
Interpreting expectation as the next outcome. It is a long-run mean.
Ignoring outcomes with zero or negative values. Include the complete distribution.
Choosing solely by expectation when risk matters. Compare variability and consequences too.
Quick self-check
- Are all possible values and probabilities included?
- Do the probabilities sum to $1$?
- Are losses represented with negative signs and costs included?
- Is the result interpreted as a long-run average?
- Does a binomial setting permit the shortcut $np$?
- Should variance, feasibility, or risk also influence the decision?
Related topics
Explore the idea
Probability sampler
Change one quantity at a time and connect what moves to Expected Value.
success other outcome
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
A game gives a net gain of $5 with probability 0.30 and a net loss of $2 with probability 0.70. Find the expected net gain in dollars.
- E[X] = 1.50 − 1.40
- E[X] = 0.10 dollar per play.
End of lesson
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