Enter up to 6 outcome values and their corresponding probabilities to compute the expected value (weighted average of outcomes). Also shows variance and standard deviation.
Expected value (EV) is the long-run average outcome of a random experiment repeated many times. It's calculated as the sum of each outcome multiplied by its probability. EV doesn't tell you what will happen in any single trial — it tells you what would happen on average over thousands or millions of trials. Casino games have negative EV for players (house edge); insurance has positive EV for insurers (premium > expected claim).
This calculator returns the expected value, variance, and standard deviation given a discrete distribution of outcomes and their probabilities. Probabilities should sum to 1.0 across all outcomes. The calculator handles up to 6 outcomes — typical for dice, simple gambling problems, and basic decision analysis.
Expected value is fundamental to: - **Gambling and casino math**: house edge, player returns. - **Insurance pricing**: premium = expected claim + costs + profit. - **Investment analysis**: expected return, risk-adjusted return. - **Decision theory**: choosing among uncertain options. - **Game theory**: optimal strategies in games.
But EV alone doesn't capture risk. A 50% chance of winning $100 and 50% of losing $80 has EV = $10 — but the variance is substantial. Most decisions consider both expected value AND variability.
**Scenario:** Win $10 on heads (P=0.5), lose $5 on tails (P=0.5). **Calculation:** EV = 0.5 × $10 + 0.5 × (-$5) = $5 - $2.50 = $2.50. SD = √(0.5 × 152.5 + 0.5 × 56.25) = √97.5 = $9.87. **Result:** EV is $2.50 per flip. Over time, you should profit. But SD of $9.87 means individual flips vary widely. Worth playing in long run.
**Scenario:** New menu item: 60% chance profit $20, 40% chance loss $15. **Calculation:** EV = 0.6 × $20 + 0.4 × (-$15) = $12 - $6 = $6. **Result:** EV = $6 per item served. Worth listing. But monitor: if rate of profitable items drops, EV becomes negative.
**Scenario:** Two-asset portfolio. Asset A: $1000 invested, 70% chance returns $1100, 30% chance returns $900. Asset B: $1000, certain return $1050. **Calculation:** A: EV = 0.7×1100 + 0.3×900 = 770+270 = $1040. B: EV = $1050 (certain). Compare EV(B) > EV(A) ($1050 > $1040). B has higher EV and no risk. **Result:** B preferred for both EV and risk reasons. A is risky AND lower expected return; no reason to choose A over B. Always look for B (or better) when available.
**Calculate expected value for:**
- **Investment decisions**: compare expected returns. - **Insurance pricing**: premium calculations. - **Game theory**: optimal strategy in repeated games. - **Casino games**: house edge calculation. - **Decision under uncertainty**: choosing best option. - **Risk-neutral analysis**: when variance doesn't matter. - **Long-run scenario planning**: expected results over time.
**EV decision rules:**
| Rule | Description | |---|---| | Maximize EV | When risk-neutral, repeated decisions | | Mean-variance | Balance EV against variance | | Utility maximization | EV of utility function | | Maximin | Maximize the worst case | | Maximax | Maximize the best case |
**When NOT to use EV alone:**
- Single critical decisions. - Catastrophic potential losses. - High variance compared to mean. - Risk-averse preferences. - Small number of decisions.
**EV and the Law of Large Numbers:**
The mean of many independent random samples converges to the expected value as n → ∞. This is why casinos always win in the long run, and insurance companies profit despite individual claims.
**EV with utility:**
For risk-averse decisions, use expected utility: E[U(X)] = Σ U(x) × P(x)
Where U is a concave utility function (diminishing returns from increasing wealth).
This explains why people buy insurance (positive expected utility despite negative EV).
**Conditional expected value:**
E(X | Y = y) = expected value of X given Y = y.
Used in Bayesian decision-making and partial information problems.
**Common applications:**
1. **Casino game**: each $1 bet returns $0.95 average. House edge 5%.
2. **Lottery**: $1 ticket has EV of ~$0.30. Pay $1, expect $0.30 long term.
3. **Insurance**: $300 premium, P(claim)=0.02, claim=$10,000. EV = -$300 + 0.02 × $10,000 = -$100. Negative EV but worth it for variance reduction.
4. **Stock market**: historical EV ~7% real annual return, SD ~16%. Most years between -25% and 39% (95% range).
5. **A/B test**: test wins by $5/visitor with P=0.7; loses by $3 with P=0.3. EV = 0.7×5 - 0.3×3 = 3.5-0.9 = $2.60/visitor. Worth running.
**Software:**
- **Excel**: SUMPRODUCT(values, probabilities). - **R**: weighted.mean() or sum(x × p). - **Python**: numpy.dot(x, p) or sum(x_i * p_i).
**Beyond simple EV:**
- **Cumulative prospect theory**: how humans actually weight outcomes. - **Stochastic dominance**: rules for comparing distributions. - **Risk-adjusted EV**: Sharpe ratio, value at risk. - **Real options**: EV of decisions with timing flexibility.
**Reporting:**
When making decisions, report: - Expected value. - Variance / standard deviation. - Range of outcomes (min, max, percentiles). - Probability of negative outcome. - Worst-case scenario.
**Long-term vs short-term:**
- Long-term: EV dominates. - Short-term: variance matters. - Number of trials: more trials = closer to EV.
**Common mistakes:**
- Treating EV as guaranteed outcome. - Ignoring variance and risk. - Using for unique critical decisions. - Failing to include all outcomes. - Confusing EV with most probable value.
Calculate basic probability from favorable outcomes divided by total outcomes.
Calculate P(X=k) and cumulative probabilities for a binomial distribution.
Calculate population and sample standard deviation from a data set.
Detect outliers in a data set using the IQR method.
Calculate box plot statistics: min, Q1, median, Q3, and max from a data set.
Generate frequency distribution from data with configurable number of bins.
Expected Value
23.0000
Variance
61.0000
Standard Deviation
7.8102
Total Probability
1.0000
Probability Check
Valid (sums to 1)
Outcomes
3