EVP
Expected Value Primer
Estimate the average long-run value of a decision by weighting each outcome by its probability. One number summarizes the bet; a second figure shows what that number does and does not promise.
Section 1 · The scenario
A decision with two outcomes
You win a small amount on heads and lose a small amount on tails.
Section 2 · Weighing the outcomes
The balance point
Expected value weights each outcome by its chance of happening and adds the results:
where p is the probability that the event happens, x₁ is the payoff if it does, and x₂ is the payoff if it does not. Picture each payoff as a weight on a beam, as heavy as its probability. The expected value is the point where the beam balances.
Probability the favorable outcome occurs.
Net gain if the favorable outcome occurs.
Net gain (or loss) if the favorable outcome does not occur.
Each payoff stands at its place on the value axis, as tall as its probability. Weighted, the first contributes +$1.50 and the second −$1, and their sum puts the balance point, marked by the triangle, on the gain side of zero: on average this decision makes money.
Section 3 · The long run
An average needs repetitions
Expected value is a promise about the average, not about any single outcome. One decision pays one payoff or the other, never the expected value itself. Repeat the decision, though, and the running total starts to track the expected value times the number of decisions made.
Faint lines are eight simulated runs; the solid line is the expected total, rising by the expected value with every decision. The shaded band, two standard deviations either side, ends between $0 and +$100, and none of the runs finished below zero. The band widens with the square root of the number of decisions while the expected total grows in proportion to it, so the average wins out only when the decision repeats.
Section 4 · Beyond the average
When the average is not enough
Two decisions with the same expected value can feel nothing alike. What separates them is spread, measured by the standard deviation of a single payoff:
For the current scenario that comes to $2.50 per decision. When the spread is large next to the expected value, or when you only get to make the decision once, the average is a poor guide to what will actually happen. That is why people buy insurance with a negative expected value: they are paying to give up the spread, not misreading the average.