The expected value of an uncertain quantity is the probability-weighted average of its possible values:
For a project whose PW might be $5,000 (probability 20%), $10,000 (probability 50%), or $15,000 (probability 30%):
EV is the simplest summary of risky outcomes, and the standard one used inside decision trees to fold back chance nodes.
Interpretation. EV is the average across many independent repetitions of the same decision. If you faced this project a thousand times with these probabilities, your average PW would be very close to $10,500. For a one-off decision the actual result is one of the three values, not the EV, but EV is still the right summary if you have no reason to weight any outcome differently.
Limits.
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Risk aversion. A decision-maker may rationally prefer a certain $8,000 over the same EV $10,500 spread across $5k to $15k; they’d rather avoid the downside. EV doesn’t capture this preference. The fix is a utility function (replace dollar payoffs with the decision-maker’s utility) and maximise expected utility instead. Beyond scope here.
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Catastrophic downside. EV treats a 1% chance of $-1 million as roughly equivalent to a 50% chance of $-20,000, since the products are the same. But a low-probability catastrophic outcome (bankruptcy, regulatory shutdown) may matter much more than a small chance of a moderate loss. Combine EV with separate analysis of the worst case, a sensitivity or stress-test approach.
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Probability estimates. EV is only as good as the probabilities. Made-up probabilities give made-up EVs. Be honest about the source of probabilities; in many engineering contexts, ranges or scenarios are more defensible than point probabilities.
Probability-free alternatives: Sensitivity analysis and Break-even analysis.