Expected value: a weighted average of possible net outcomes

Start with the outcomes and the unit

To calculate an expected value, multiply each possible result by its probability, then add the products. You get a probability-weighted average in the same unit as the results. Before you calculate, decide what you are measuring: total time, time saved, or time saved after preparation. Those are different quantities.

Imagine a task that otherwise takes 60 minutes. A different method needs 12 minutes of preparation on every attempt, followed by 20, 45 or 75 minutes of work. We assign those three cases probabilities of 1/4, 1/2 and 1/4. These are invented times and assumptions, not observations about a real task. After preparation, the possible savings are 28, 3 and −27 minutes; a negative saving means extra time.

The question is how to combine those unequal chances into one expected net saving. The calculation gives 1.75 minutes. It does not say that your next attempt will save 1.75 minutes, and it does not supply evidence that the assumed probabilities are accurate.

Validate the probability table

The rows must describe cases that do not overlap and together cover every outcome in the model. Each probability must be at least zero, and all probabilities must sum to 1. Here, exactly one of the three work times occurs. The weights add up: 1/4 + 1/2 + 1/4 = 1. Calling these cases independent would be wrong: if one occurs, the other two do not occur on that attempt.

If a real task has other possible times, this three-row table would not describe it completely. You would need to revise the cases and justify their probabilities. Dividing an incomplete set of weights by its total does not recover an omitted outcome.

Also check the denominator. In a separate hypothetical set of 100 equally likely cases, suppose B occurs in 40 and both A and B occur in 12. The joint probability of A and B is 12/100 = 12%; the conditional probability of A given B is 12/40 = 30%, because it uses only the B cases. Conditioning on B this way requires P(B) > 0. These figures do not assume independence or replace the work-time weights above. To calculate an expected value only within the B cases, use a complete set of probabilities conditional on B, not a mixture of conditional and whole-population weights. See the University of Arizona probability notes, §1.2 for the conditional-probability definition; the 100-case numbers here are invented.

Subtract preparation once, then weight the net results

Let X be the net minutes saved on one attempt. The expected value is E[X] = Σ pᵢxᵢ: add each result xᵢ multiplied by its probability pᵢ. In this example, E[X] = (1/4 × 28) + (1/2 × 3) + (1/4 × −27) = 7 + 1.5 − 6.75 = 1.75 minutes.

Before preparation, the expected saving is 60 − 46.25 = 13.75 minutes. Subtracting 12 minutes gives the same net value of 1.75 minutes. If you have already used 28, 3 and −27 as the net outcomes, do not subtract 12 again. Also, taking the simple average of those three net outcomes would give 4/3 minutes, which ignores the middle case's larger weight.

The expectation rule for subtracting costs does not require the work time and the cost to be independent. Here the preparation is a fixed 12 minutes. If preparation instead differs by case, put the appropriate preparation time into each row before calculating the weighted net result. Keep the units consistent throughout.

Separate the mean, a possible result and the chance of a positive result

The expected net saving is 1.75 minutes, but none of the three cases saves that amount. An attempt saves 28 or 3 minutes, or takes 27 minutes longer than the baseline. The chance of a positive saving is 3/4, because the first two cases save time. That probability is a different quantity from the expected saving: one is a chance, the other is measured in minutes.

An average calculated from completed attempts is another quantity. It depends on what actually happened in that set of attempts. The weighted mean here comes from the stipulated model; it does not promise that any particular finite set of observed results will have an average of 1.75 minutes.

Change preparation time while holding the other assumptions fixed

Let c be preparation time in minutes on every attempt. Keep the three work times and their original probabilities. The expected net saving becomes 13.75 − c minutes. This isolates one assumption: changing preparation time changes each case's net saving, while the work-time distribution stays the same.

The 13.75-minute threshold is an algebraic result under these assumptions. It is not a measured estimate of preparation time or a guarantee of breaking even on an attempt. If the work times or their probabilities also change, this particular threshold no longer describes the revised model.

Change probabilities without losing a valid distribution

Return preparation to 12 minutes. Keep the middle case's probability at 1/2, call the long case's probability q, and give the short case probability 1/2 − q. Increasing q now moves probability from the short case to the long case. For 0 ≤ q ≤ 1/2, all three weights are nonnegative and still sum to 1.

The resulting expected net saving is 28(1/2 − q) + 3(1/2) − 27q = 15.5 − 55q minutes. It is zero at q = 31/110. That fraction is the model's switching point, not an estimate of how often the long case really happens.

These are alternative assumptions, not measured probabilities or a claimed plausible range. Sensitivity calculations show how the answer depends on an input; they cannot establish that input's accuracy. A real application would need evidence for its cases, times and probabilities, including outcomes the initial model may have missed.

A positive expectation still permits a worse result

In the original model, the 75-minute work case takes 87 minutes including preparation, 27 minutes longer than the baseline. Its probability remains 1/4 even though the expected net saving is positive. A single expected value does not describe the full range of outcomes.

Nor does a positive expectation guarantee a net saving after a fixed number of attempts. For example, under the additional assumption of independent attempts, a finite run consisting entirely of the long case still has positive probability. This is a limit of the claim, not a reason to repeat an activity until a loss is recovered.

Common questions

Can the expected value be a result that never occurs?

Yes. A weighted average need not be one of the possible results. Here, 1.75 minutes is the expected net saving, while an attempt gives 28, 3 or −27 minutes. Keep the list of possible outcomes beside the mean so you can see what the calculation leaves out.

Is a 75% chance the same as an expected saving of 1.75 minutes?

No. The 75% figure adds the probabilities of the two positive outcomes. The expected saving weights all three amounts, including the negative one. A probability has no time unit; the expected saving is measured in minutes.

Do I subtract preparation before or after weighting?

Either route works when applied consistently. Subtract the fixed 12 minutes from each case before weighting, or subtract it once from the expected saving before preparation. Do not subtract it from the net outcomes and then subtract it again from their expected value.

What if the probabilities are only assumptions?

The result is conditional on those assumptions. Label them, keep the weights valid, and show how changing them affects the answer. That calculation does not turn assumed probabilities into observed chances or guarantee a real result.

Sources and scope

The numerical task is fictional. Its times and probabilities were chosen for this explanation. The sources below support the mathematical definitions and method; they do not validate the example as real performance data.

1. OpenStax, Statistics §4.2: expected value as a probability-weighted mean.
2. Penn State STAT414, Lesson 7: discrete outcomes and valid probability mass functions.
3. Penn State STAT414, Lesson 8: expectation and transformations, including the distinction between a mean and a realized value.
4. Northeastern University course notes, Random Variables §2.1: finite expectation, linearity and qualified independence rules.
5. HM Treasury, The Green Book (2026): sensitivity and switching-value methods in UK public appraisal. This is a limited methodological reference, not a worldwide financial or legal rule.

References checked: 17 October 2026. This is the date of consultation, not the sources' publication date. The Green Book page shows an update date of 5 February 2026.