How to Calculate Expected Value

Expected value is a weighted average. Each outcome contributes its value multiplied by how likely it is, and you sum the contributions.

For a binary case: the probability of the favourable outcome times what you gain, minus the probability of the unfavourable outcome times what you lose. If those two probabilities sum to one, that is the whole calculation.

For more outcomes, the same operation with more terms. Nothing about the arithmetic is subtle, and any spreadsheet performs it.

What makes the number meaningful or meaningless is the probability you supplied.

That probability is an estimate. It comes from a model, a historical base rate, a judgement, or a market price, and the calculation cannot tell the difference between a well-founded estimate and a guess. It will produce a confident output from either.

This is why expected value calculations proliferate and reliable ones do not. The visible part is arithmetic and the invisible part is an estimate nobody audits, so the output inherits a precision it does not have.

The useful discipline is to treat the probability as the deliverable and the arithmetic as a formality.

A model. Some process that maps inputs to a probability. This is the only source that can be systematically evaluated, because it makes repeated claims you can score against what happened. It is also the only source that can be wrong in a consistent, correctable direction.

A base rate. How often this kind of thing has happened historically. Straightforward, and it assumes the conditions that produced the history still hold, which in sport is frequently untrue after a rule change, a roster change or a scheduling change.

Market-implied. Derived from a posted price. This is the trap. If you take an implied probability from the price and then compute expected value against that same price, the answer is negative by exactly the margin, always, because you have assumed the market is right and then measured yourself against it. Any apparent edge in that setup is an arithmetic artefact.

The only non-circular use of market prices is as a comparison. Your estimate from an independent source, compared against the de-margined market estimate, produces a difference that may mean something. Using the market as your input and the market as your benchmark produces nothing.

Judgement. Legitimate as an input and untestable in isolation. If used, write it down before the outcome, because unrecorded judgement is reconstructed after the fact in a systematically flattering way.

Sensitivity is more informative than the point estimate

It describes a long-run average. Expected value is what the average outcome converges toward over many independent repetitions. It says nothing about any individual instance, and in domains with high variance and few repetitions the average may not be approached within any horizon you care about.

It ignores variance entirely. Two situations with identical expected value and very different spreads are not equivalent in any practical sense, and the formula cannot see the difference. Reporting expected value without the distribution around it omits most of the relevant information.

It is only as good as its input, and the input is unaudited. This bears repeating because the arithmetic's precision is misleading. A figure computed to several decimal places from a probability someone guessed is a guess with decimal places.

It assumes the estimate is unbiased. If a model is systematically optimistic, every calculation built on it is systematically optimistic, and the error does not average out because it is not random.

For practical work: record the probability, its source, and the method used to produce it alongside every calculation. Recompute historical figures when the method changes. And evaluate the probability estimates themselves against what happened, since that is the only part of the chain that can actually be validated.

This page describes a calculation and is not betting advice.