- Calculate a weighted average of possible payoffs.
- Use linearity even when terms are dependent.
- Break counting problems into indicator variables.
Concepts
Expectation is a weighted average
For a discrete variable, when the expectation exists. It describes a model average, not a promise for the next outcome and not necessarily a value X can take. Keep losses negative and distinguish a gross payout from a net payoff after fees.
Linearity is the workhorse
for variables with finite expectations. Independence is not required. This often avoids finding the entire distribution of a complicated total. In contrast, is not generally valid without additional assumptions such as independence.
Count with indicators
An indicator I is 1 when an event occurs and 0 otherwise, so E[I]=P(event). If a total counts how many events occur, write it as a sum of indicators and add their probabilities. You can also average conditional means: split the model into exhaustive cases and weight each conditional expectation by its case probability.
Examples
Worked example 1
A signed payoff
A game gives a net payoff of +8 units with probability 1/4 and −2 units otherwise. What is its expected net payoff?
List all net outcomes
The payoffs already include any costs, so use +8 and −2 directly. Their probabilities are 1/4 and 3/4.Weight and add
.Interpret the answer
The model average is +0.5 units per game. No individual game pays 0.5, and a positive mean does not remove risk.
Worked example 2
Correctly returned badges
Ten named badges are randomly permuted among their ten owners, with every permutation equally likely. How many owners get their own badge, on average?
Use one indicator per owner
Let I_i be 1 if owner i receives the correct badge. The total correct returns is X=I_1+⋯+I_10.Use symmetry for one owner
Each owner is equally likely to receive any of ten badges, so E[I_i]=P(I_i=1)=1/10.Add without requiring independence
E[X]=10×(1/10)=1. Correct-return events are dependent, but linearity still applies.
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Expectation is a balance point
Place probability weight at each possible payoff. The mean balances those weights and stays between the outcomes; it need not be a payoff you can receive.
Expected payoff0.50 units
75.0% × -2 + 25.0% × 8. A positive expectation describes a model average, not a guaranteed gain.
| Payoff | Probability | Weighted contribution |
|---|---|---|
| -2 | 75.0% | -1.50 |
| 8 | 25.0% | 2.00 |
Concept checks
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Key results
- Keep payoff signs and fees consistent.
- Add expectations before seeking a full distribution.
- Write counts as sums of indicators.
- Average conditional means with the correct weights.
Practice problems
Solve related problems or try a timed probability drill.