Expected value is the arithmetic for comparing uncertain choices. Multiply each possible outcome by the chance of it happening, add the results, and you have a single number that can be set against the number from another option.

A 20% chance of winning a $50,000 project has an expected value of $10,000. A 70% chance of a $12,000 project has an expected value of $8,400. The first is worth more, despite being much less likely, and without the arithmetic it is genuinely hard to say which is better — which is what the tool is for.

What it makes visible

The cost of pursuit is a real cost. If the $50,000 pitch takes three days of work you could have billed, and its expected value is $10,000, the comparison is between $10,000 and three billed days plus the next-best use of them. Most businesses bid on the headline figure and discover the arithmetic by accident, in a quarter where they lost several large pitches and did no billable work.

Longshots can be correct. A 10% chance at something very large can beat a 90% chance at something small. This is the defensible core of taking a swing, and it is also why the version of it that is not defensible — the 2% chance at something large, repeated until the money runs out — looks superficially like the same argument.

Small edges are worth having. A decision made with a 5% advantage, repeated a hundred times, reliably beats one made without it. Most operational improvements are of this kind and feel too small to bother with, which is exactly why they persist as opportunities.

The probabilities are the hard part

Nobody hands you the numbers. In a business decision the probability is an estimate, and the arithmetic is only as good as it.

The protection against inventing them is the base rate — start from how often this sort of thing has worked before, for you or for anyone, and adjust from there. A firm that wins one pitch in six should not be estimating 50% on the current one because it feels promising. Feeling promising is not evidence, and it is the most reliably overweighted input in the whole exercise.

Writing the estimate down before the outcome is known is the only way to find out whether your estimates are any good. Almost nobody does this, and almost everybody believes their judgment is well calibrated.

Where it stops working

Expected value is the average over many repetitions. When the decision is not repeated, or cannot be survived, the average is the wrong guide.

A bet with a positive expected value and a 5% chance of losing everything is a good bet if you can take it a thousand times and a terrible one if you take it once, because the 5% ends the sequence. The average across many players is not the average across one player's life — this is ergodicity, and it is the single most important qualification on the whole method.

Which is why survival gets treated as a constraint rather than as one more outcome to be weighted. Rule out the branches that end the business, then maximize expected value among what is left. A margin of safety is what that constraint looks like in practice.

The other limit is timing: expected value says nothing about when. A probability-weighted payment three years away is worth less than the same one today, and comparing them requires discounting as well as weighting. Doing one and not the other is how a deferred, uncertain sum gets talked about as though it were cash.