Expected Value Explained: How to Weigh Uncertain Outcomes
Expected value is the long-run average of a choice, weighted by how likely each outcome is. It is the first tool for comparing risky options, and its limits matter as much as its arithmetic.
The definition
The expected value of a random quantity is, informally, the average result you would get if you could repeat the same situation many times. To calculate it, multiply each possible outcome by its probability and add the results.
E = 0.6 × 10 + 0.4 × 0 = 6 unitsNo single attempt returns 6; each returns 10 or 0. Six is what the attempts average out to over many tries.
Expected value in mission planning
Agencies and companies use the same reasoning, with much more careful numbers, to compare options. Should a mission carry a spare instrument, or more propellant? Should a launch wait a day for better weather? Each choice changes the probabilities of different outcomes and their costs, and expected value is one way to compare them on a common scale.
Why expected value is not the whole story
Two options can have the same expected value and very different risks. Variance measures how spread out the possible outcomes are. A guaranteed 6 units and a 60% chance of 10 have the same expected value, but the second can leave you with nothing. For one-off, irreversible events, such as a crewed launch, planners care a great deal about the bad tail of the distribution, not just the average.
- Repeatable bets can be judged mainly by expected value, because the average shows up over many trials.
- One-off bets need attention to the worst case and to whether you can survive it.
- Uncertain probabilities make every calculation shakier; small errors in a probability can swing the result.
Offers with conditions
Everyday life is full of offers whose headline value differs from their expected value, because conditions change what you actually receive: a trial credit that must be used a number of times before it can be withdrawn, or a discount that applies only above a minimum spend. The honest way to value such an offer is to write down every condition, estimate the outcomes and their probabilities, and only then compute the average. A worked example of how this framing shows up in practice is a Traditional Chinese guide that walks through casino trial bonuses, how they are claimed and the wagering conditions attached to them: a 2026 overview of online casino trial bonuses and their conditions (娛樂城體驗金懶人包). Read any page like that with this article's arithmetic in mind: in games where the house has an edge, every extra round you are required to play lowers the expected value of what is left.
A short checklist
- List every outcome, including the boring ones.
- Give each a probability, and note how confident you are in it.
- Multiply and add for the expected value.
- Then look at the worst case separately.
Why does the average "show up" over many trials at all? That is the law of large numbers.
REV 2026-11-09 / Tobias Wren