Expected Value CalculatorFree online math calculator — instant results, no sign-up
Calculate an expected value from outcomes and their probabilities.
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About the Expected Value Calculator
The Expected Value Calculator pairs each outcome in Values with its probability in Paired values or weights to calculate a probability-weighted average. The default entries 2, 4, 6, 8, 10 and 1, 2, 3, 4, 5 produce “Check your inputs” rather than an expected value because the second list is not a valid probability distribution.
- Values
- Enter the numerical outcomes in matching order, all expressed in the same units, such as counts or currency.
- Paired values or weights
- For an expected-value calculation, enter the corresponding unitless probabilities, each between 0 and 1 and together totaling 1.
Formula and method
E[X] = Σᵢ₌₁ⁿ xᵢpᵢ, subject to 0 ≤ pᵢ ≤ 1 and Σᵢ₌₁ⁿ pᵢ = 1. For the default inputs, Σpᵢ = 1 + 2 + 3 + 4 + 5 = 15, so the probability conditions fail and Result = Check your inputs.
- XThe numerical random variable whose possible outcomes are entered.
- E[X]The expected value, expressed in the same units as the outcomes.
- xᵢThe outcome at position i in Values.
- pᵢThe probability paired with the outcome at position i.
- nThe number of outcome-probability pairs.
- iThe index identifying a matching pair, from 1 through n.
- ΣSummation over all matching pairs.
Match the two lists by position and check that they contain equal numbers of valid numerical entries. A discrete expected-value calculation requires nonnegative probabilities no greater than 1, with a total of 1. When those conditions hold, multiply each outcome by its probability and add the products. With the supplied defaults, the five entries match in count, but the probability list totals 15 and includes entries above 1; the displayed result is therefore “Check your inputs,” not a numerical expected value.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Values2, 4, 6, 8, 10
- Paired values or weights1, 2, 3, 4, 5
Results
- ResultCheck your inputs
How to use the Expected Value Calculator
- 1Enter the outcomes in Values; the default list is 2, 4, 6, 8, 10.
- 2Enter the corresponding probabilities in Paired values or weights, preserving the same order.
- 3Confirm that both lists have the same number of entries and that the probabilities lie between 0 and 1 and total 1.
- 4Run the calculation; leaving the second list at its default of 1, 2, 3, 4, 5 produces “Check your inputs.”
- 5If validation fails, correct the probability entries and calculate again; read a valid numerical result in the units of Values.
Reading your result
The provided example has one output row: Result. Its value, “Check your inputs,” is a validation message, not zero or an expected value. For valid probability inputs, a numerical Result represents the probability-weighted mean of the listed outcomes, not necessarily an outcome that can occur on a single trial.
Limitations
The expected-value formula assumes a finite list of outcomes with probabilities covering the full distribution. Negative outcomes are allowed, but negative probabilities are not. Although the second field mentions weights, the supplied example does not demonstrate automatic weight normalization, so do not assume arbitrary weights are accepted directly. The example also does not specify the calculator's rounding tolerance for probability totals. Expected value alone does not describe variability, risk, or the most likely outcome.
Common questions
Why do the default inputs return “Check your inputs”?+
The outcomes 2, 4, 6, 8, 10 are numerical, but their paired entries 1, 2, 3, 4, 5 are not valid probabilities: they total 15, and several exceed 1.
Can I use the default second list as relative weights?+
If 1, 2, 3, 4, 5 are intended as relative weights, convert them to probabilities by dividing each by their total of 15 before entering them. This changes the inputs; the unchanged defaults still produce “Check your inputs.”
Does the order of the values and probabilities lists matter for expected value?+
Yes. With the defaults, 2 is paired with 1, 4 with 2, 6 with 3, 8 with 4, and 10 with 5. Reordering complete pairs preserves an expected value, but reordering only one list generally changes it.
Must the expected value equal one of the outcomes in Values?+
No. With valid probabilities, it is a weighted average and may fall between listed outcomes. It must lie between the smallest and largest outcomes.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
Further reading
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Results are estimates for informational purposes and are not financial, medical or legal advice.