Expected value
Brainteasers & GuesstimatesThe probability weighted average of every possible outcome, and the standard basis for choosing between uncertain payoffs.
Also written: probability weighted average, expected profit
Expected value multiplies each outcome by its probability and adds the results. It is the number a decision converges on if you could run it many times, which is exactly the condition under which it is the right rule and exactly the condition people forget to check.
The mechanical point that trips candidates up is which figures get weighted. Weight what is uncertain and take what is certain at face value. A contingent fee is multiplied by its probability. A cost you incur whether you win or lose comes off in full, because you pay it in every branch of the tree.
Two conditions have to hold before expected value should decide anything. The decision has to repeat often enough for the average to arrive, and you have to survive every individual outcome. A bank choosing which pitches to run satisfies both. A single position with a positive expectation and a small chance of a loss that ends the firm satisfies neither, which is why risk limits exist alongside expected return and why practitioners argue about how much variance a positive expectation is allowed to carry.
The last trap is a payoff that is not linear in the outcome. Where the payout depends on the square of a draw, the expected payout is not the payout at the expected draw, and the gap is the variance. Using the average input in a nonlinear payoff is an error, not a rounding.
Worked example
A pitch has a 30% chance of winning a €4m fee, and costs €300,000 to pursue whether or not it is won.
The expected fee is 30% of €4m, which is €1.2m. The €300,000 is certain, so it comes off in full: expected profit €900,000.
Weighting the cost by 30% as well gives €1.11m. That answer assumes you only pay for the pitches you win, which is not how pitching works, and it is the standard error on this question.