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Probability

Discrete Expected Value Calculator

Enter four discrete outcomes and their probabilities to reconcile the distribution's expected value, second moment, variance, standard deviation, and difference from a comparison value. The guide shows why probabilities must sum to one and how outcome units propagate.

Outcome 1Probability 1Outcome 2Probability 2Outcome 3Probability 3Outcome 4Probability 4Entered comparison value

Entered-distribution expected value-

Total entered probability-

Outcome 1 contribution-

Outcome 2 contribution-

Outcome 3 contribution-

Outcome 4 contribution-

Probability-weighted squared outcome-

Variance from second moment-

Distribution standard deviation-

Expected value minus comparison-

Decision view

Discrete probability mass and expected-value balance

Discrete probability mass and expected-value balanceOutcome lollipops are sized by probability while weighted contributions reconcile to the expected center and distribution spread.

Exact scenario comparisonProbability 4 changes while all other entered assumptions remain constant.

Probability 4	Entered-distribution expected value	Total entered probability	Outcome 1 contribution	Outcome 2 contribution	Outcome 3 contribution	Outcome 4 contribution	Probability-weighted squared outcome	Variance from second moment	Distribution standard deviation	Expected value minus comparison

How to use Discrete Expected Value Calculator

Enter four mutually exclusive outcomes in one consistent unit.
Assign a probability between zero and one to each outcome.
Confirm the displayed probability total equals one.
Read contribution, expectation, and spread as separate properties of the distribution.

Calculator guide

Understanding Discrete Expected Value Calculator

Expected value is the probability-weighted center of a discrete distribution, not necessarily one of its possible outcomes. This calculator exposes every weighted contribution, the probability total, the second moment, variance, and standard deviation.

Mass must reconcile A complete distribution totals one.

Center may be unattainable Expectation need not equal a listed outcome.

Units matter Variance is squared while SD returns to original units.

Drivers stay visible Every p_i x_i contribution is shown.

Calculation method

How the calculation works

Multiply four outcomes by their entered probabilities, sum contributions, and derive variance from the second moment while displaying the probability total. In the Discrete Expected Value Calculator, the live scenario varies probability 4 and tracks entered-distribution expected value while the remaining results preserve the reconciliation path. Multiply each outcome by its probability and add the four contributions. Multiply each squared outcome by its probability for the second moment, then subtract the square of the expected value to obtain variance.

Detailed calculation process

Weight each possible outcome before measuring spread

The default distribution assigns probabilities 0.15, 0.35, 0.30, and 0.20 to outcomes 0, 20, 50, and 100.

General formula: E[X] = sum(p_i x_i); E[X^2] = sum(p_i x_i^2); Var(X) = E[X^2] - E[X]^2; SD(X) = sqrt[Var(X)] Probabilities act as long-run shares. The first moment locates the center, while the second moment retains squared distance information needed to derive variance.

What each symbol means

x_i Possible outcome i in the outcome's original unit.

p_i Probability assigned to outcome i, unitless and between 0 and 1.

E[X] Expected outcome in the same unit as x.

E[X^2] Second moment in squared outcome units.

Var(X) Distribution variance in squared outcome units.

SD(X) Distribution standard deviation in the original outcome unit.

Worked substitution with the default inputs

1. Reconcile probability mass: 0.15 + 0.35 + 0.30 + 0.20 = 1.00 A total of one makes the entries a complete discrete probability distribution.

2. Calculate weighted contributions: 0x0.15 = 0; 20x0.35 = 7; 50x0.30 = 15; 100x0.20 = 20 The high outcome contributes strongly despite having only 20% probability.

3. Add the expected value: E[X] = 0 + 7 + 15 + 20 = 42 The long-run average is 42 outcome units even though 42 is not an entered outcome.

4. Calculate the second moment: E[X^2] = 0 + 20^2x0.35 + 50^2x0.30 + 100^2x0.20 = 2,890 Squaring gives distant outcomes greater influence on spread.

5. Recover variance and standard deviation: Var(X) = 2,890 - 42^2 = 1,126; SD = sqrt(1,126) = 33.5559; 42 - 45 = -3 Variance is in squared units; standard deviation returns to the outcome unit.

The default probabilities sum to 1, the expected value is 42, variance is 1,126, standard deviation is about 33.556, and the expectation is 3 below the entered comparison.

Distribution audit

Read the shape behind the average

Two distributions can share an expected value while having very different risk.

Probability total Checks whether the distribution is complete.

Contribution mix Shows which outcomes drive the center.

Second moment Captures the leverage of distant outcomes.

Standard deviation Summarizes spread in outcome units.

Worked situations

Practical examples

An outcome of 100 with probability 0.20 contributes 20 to expected value.
The expected value 42 is a long-run average, not an additional possible outcome.
A wide outcome range produces a standard deviation of about 33.556.

Better inputs

Useful tips

Keep all probabilities on the same decimal scale.
Include zero-probability outcomes only when they help document the model.
Use contribution size to see which scenarios drive the expectation.

Before relying on the result

Limitations and common mistakes

Probabilities that do not sum to one do not define a complete distribution.
Expected value does not describe tail risk, utility, or path dependence by itself.
Estimated probabilities and outcomes can change, and this arithmetic does not quantify their estimation error.

Reference

Key terms

Expected value: Probability-weighted average across possible outcomes.
Probability mass: The probability assigned to a discrete outcome.
Second moment: Probability-weighted average of squared outcomes.

Important note

Calculated directly from the entered values using the displayed formula and rounding settings.

Frequently asked questions

Can expected value be an impossible outcome?

Yes. It is a weighted average across repetitions, not necessarily a value attainable in one trial.

Why must probabilities sum to one?

Because the listed mutually exclusive outcomes must account for all probability mass in a complete distribution.

Why square outcomes for variance?

The second moment retains distance information without positive and negative deviations canceling.

Can I enter percentages such as 20 instead of 0.20?

No. Enter probabilities on the 0-to-1 decimal scale.