Decisions
20 min

Expected value

Multiply each outcome by its probability and sum, to get the average result of a choice repeated many times.

Time cost
20 min
Output
An EV per option with the probabilities stated.
Steps
5
Run it — Expected value and decision tree →

Use when

  • Outcomes are quantifiable and probabilities can be estimated within an order of magnitude.
  • The decision repeats, or is one of many similar decisions.
  • A loss is survivable, so the average is a meaningful guide.

Do not use when

  • One branch is ruin. The average is irrelevant if you cannot play again.
  • The decision happens once and the variance is large relative to what you hold.
  • Probabilities are pure invention. Precise arithmetic on invented inputs produces confident nonsense.

Inputs required

  • Outcomes in a common unit
  • A probability per outcome, summing to 1

Procedure

  1. 01

    Enumerate outcomes

    List the distinct results, not the ranges. Make them mutually exclusive and exhaustive, so the probabilities can legitimately sum to one.

  2. 02

    Assign probabilities

    Use a base rate where one exists. Check they sum to 1 — if they do not, an outcome is missing or double-counted.

  3. 03

    Value each outcome

    In one unit. Include the costs you would actually bear, not only the headline number.

  4. 04

    Multiply and sum

    EV is the sum of probability × value. Compare across options.

  5. 05

    Check the spread before acting

    Two options with the same EV can have very different ranges. Look at the worst branch and ask whether you survive it. If not, EV is the wrong tool and you want the risk cluster.

Characteristic failure mode

False precision. Probabilities estimated to two decimals from nothing produce an EV with the authority of arithmetic and the content of a guess. Round to the nearest 5% and see whether the conclusion still holds.

Worked example

Choosing whether to pay £400 for an extended warranty on a £2,000 appliance.

  1. 01Base rate for a major failure in the covered period: roughly 6%.
  2. 02Cost of a failure without cover: about £900.
  3. 03EV of not buying: 0.06 × −£900 = −£54.
  4. 04EV of buying: −£400 plus residual excess.

Result

Not buying is better by roughly £340 in expectation. It is also survivable, which is what makes the expectation the right basis.

Where this disagrees with another method

vs. Minimax regret →

EV optimises the average; minimax regret optimises the worst case. Trust EV when the decision repeats and losses are survivable. Trust minimax regret when it happens once and you must live with the worst branch.

Where to go next

Also cited by
Minimax regret