# Expected value calculator

> Calculate expected value E[X], variance and standard deviation for a bet, game or decision from its outcomes and their probabilities or weights.

Interactive version: https://www.calcopenly.com/statistics/expected-value-calculator
Subject: Statistics and probability calculators

Expected value is the probability-weighted average of the outcomes, E[X] = Σ xᵢpᵢ: what a bet, game or decision returns per play over many plays. The variance, Σ(xᵢ − E[X])²pᵢ, and its square root, the standard deviation, measure how far single results swing around that average. Counts or relative weights are rescaled to probabilities before the sums.

The default is a $5 scratch card that loses the stake with probability 0.8 and wins a net $5, $20 or $95 with probabilities 0.15, 0.04 and 0.01. Its expected value is −$1.50 per card, even though 20% of cards come out ahead.

Enter each outcome as a net result, winnings minus the stake: a lost $5 stake is −5 and a $100 prize on a $5 card is 95. The probabilities must add up to 1, or switch to counts and weights.

## Inputs

- **Outcomes (values or payoffs)**: Net result of each outcome, e.g. −5 for losing a $5 stake.
- **Probabilities**: One per outcome, in the same order. Fractions such as 1/38 work.
- **Second list holds** (options: Probabilities (sum to 1), Counts or weights)

## Results

- Expected value E[X] — main result
- Variance
- Standard deviation
- Probability of a positive outcome
- Worst outcome
- Best outcome

## Formula

$$
E[X] = \sum_i x_i\,p_i,\qquad \operatorname{Var}(X) = \sum_i (x_i - E[X])^2\,p_i
$$

## Worked examples

### $5 scratch card (defaults)

- Outcomes (values or payoffs): -5, 5, 20, 95
- Probabilities: 0.8, 0.15, 0.04, 0.01
- Second list holds: Probabilities (sum to 1)
- **Expected value E[X]: -1.5**
- **Variance: 127.75**
- **Probability of a positive outcome: 0.2**
- Checked against: Hand calculation: −4 + 0.75 + 0.8 + 0.95 = −1.5; Σp(x − μ)² with Python fractions = 127.75

### American roulette, single-number bet

- Outcomes (values or payoffs): 35, -1
- Probabilities: 1/38, 37/38
- Second list holds: Probabilities (sum to 1)
- **Expected value E[X]: -0.052632**
- Checked against: House edge of a straight-up bet: −2/38 = −5.26% (standard roulette tables)

### Fair die

- Outcomes (values or payoffs): 1 2 3 4 5 6
- Probabilities: 1 1 1 1 1 1
- Second list holds: Counts or weights
- **Expected value E[X]: 3.5**
- **Variance: 2.916667**
- Checked against: E = 7/2 and Var = 35/12 for a fair die (textbook result; Python fractions)

### Edge case: a certain outcome

- Outcomes (values or payoffs): 42
- Probabilities: 1
- Second list holds: Probabilities (sum to 1)
- **Expected value E[X]: 42**
- **Variance: 0**
- **Standard deviation: 0**
- Checked against: A single outcome with probability 1 has no spread

## Questions

### How do you calculate expected value?

Multiply each outcome by its probability and add the products: E[X] = Σ xᵢpᵢ. For a fair six-sided die, E = (1 + 2 + 3 + 4 + 5 + 6) × 1/6 = 3.5. For the default scratch card, −5 × 0.8 + 5 × 0.15 + 20 × 0.04 + 95 × 0.01 = −1.5, a loss of $1.50 per $5 card on average.

### What does a negative expected value mean?

The bet loses money on average per play. The −$1.50 scratch card returns −30% of its $5 price, so 100 cards are expected to lose $150. Single results vary: with a standard deviation of 11.30 per card, the total over 100 independent cards has a standard deviation of 11.30 × √100 = 113, so some buyers of 100 cards still come out ahead.

### What is the house edge in roulette?

5.26% on an American double-zero wheel and 2.70% on a European single-zero wheel. A single-number bet pays 35 to 1. With 38 pockets, the expected value per unit staked is (35 − 37)/38 = −2/38 = −0.0526; with 37 pockets it is (35 − 36)/37 = −1/37 = −0.0270. Almost every other American bet has the same 5.26% edge.

### Is expected value the most likely outcome?

No. It is a long-run average and may not be a possible result at all: a die's expected value is 3.5, which no roll shows. The scratch card's expected value is −1.5, but its most likely outcome is losing the full $5, which happens 80% of the time. In a skewed payoff like this, a few large prizes pull the average above the typical result.

### Is the option with the highest expected value always the best choice?

Not always, because expected value ignores risk. A sure $50 and a 50% chance of $100 both have an expected value of 50, but standard deviations of 0 and 50. Insurance has a negative expected value for the buyer yet is rational when it removes a loss the buyer could not absorb. Expected utility theory (von Neumann and Morgenstern, 1944) formalizes this trade-off.

### How accurate is the expected value calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 4 worked examples whose answers come from independent sources; for example, “$5 scratch card (defaults)” is checked against Hand calculation: −4 + 0.75 + 0.8 + 0.95 = −1.5; Σp(x − μ)² with Python fractions = 127.75.

### Where does the method come from?

Grinstead & Snell, Introduction to Probability (AMS), chapter 6 Expected value and variance; NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.1 What is a probability distribution.

## Sources

- [Grinstead & Snell, Introduction to Probability (AMS), chapter 6 Expected value and variance](https://math.dartmouth.edu/~prob/prob/prob.pdf)
- [NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.1 What is a probability distribution](https://www.itl.nist.gov/div898/handbook/eda/section3/eda361.htm)