How to use this calculator
- Enter or choose outcomes (values).
- Enter or choose probabilities.
- Enter or choose probabilities are.
- The expected value E[X] updates as you type. Use Copy result or Share to keep it.
Method
E[X] = Σ xᵢpᵢ; Var(X) = Σ pᵢ(xᵢ − E[X])²; SD = √Var(X)
- xᵢ
- each outcome
- pᵢ
- its probability
Worked example
- E[X] = (1 + 2 + 3 + 4 + 5 + 6) ÷ 6 = 21 ÷ 6 = 3.5.
- Var = (6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25) ÷ 6 = 17.5 ÷ 6 = 2.916667; SD = 1.7078.
A fair die has an expected value of 3.5 with a standard deviation of about 1.71.
What the result means
A negative expected value for a bet means you lose on average in the long run. The standard deviation shows how far single results typically stray from that average.
Assumptions and limits
- Outcomes are mutually exclusive and the probabilities cover every possibility.
- Expected value ignores risk preferences: two choices with the same EV can differ greatly in how much you could lose.
Questions
Does the expected value have to be a possible outcome?
No. A die never shows 3.5; it is a long-run average.
How do I value a lottery ticket?
List each prize (net of the ticket price) with its probability, including the “lose the stake” outcome; the EV is almost always negative.