Expected Value Calculator

About the Expected Value Calculator

The Expected Value Calculator computes the weighted average outcome of a probability distribution — one of the most important concepts in statistics, gambling, finance, and decision-making. Expected value (EV) tells you what you'd average over many repetitions of a random event, so it is the right starting point for evaluating bets, projects, and uncertain choices. It helps turn a list of possible outcomes into one comparable long-run number.

Beyond the mean, this calculator also computes variance and standard deviation to quantify the spread of possible outcomes. A positive EV with low variance means a reliably good bet; a positive EV with high variance means wild swings despite long-term profit. Both metrics are essential for rational decision-making under uncertainty.

Enter outcome values and their corresponding probabilities to see the expected value, risk metrics, and a visual breakdown of the probability distribution. Presets for common scenarios like dice rolls, lottery tickets, and investment decisions make exploration easy. It is a fast way to turn uncertainty into a number you can compare.

When This Page Helps

Use this calculator when you need a weighted average outcome instead of a single best-case or worst-case scenario. It is helpful for comparing uncertain choices, checking whether a payoff structure is favorable on average, and pairing the mean with variance so the risk is visible too before you commit to a decision. That makes the upside and downside easier to judge together.

How to Use the Inputs

1. Enter outcome values (payoffs, returns, or any numeric results) in the value fields.
2. Enter the corresponding probability of each outcome (must sum to 1.0 or 100%).
3. Add more rows for additional outcomes using the Add Outcome button.
4. Or select a preset scenario to auto-fill common distributions.
5. Review the expected value, variance, standard deviation, and coefficient of variation.
6. Examine the probability distribution chart for visual insight.
7. Compare different scenarios by adjusting probabilities and values.

Example Calculation

Tips & Best Practices

• Always verify probabilities sum to 1.0 — extra or missing probability invalidates results.
• Use coefficient of variation (CV) to compare risk across different scales of outcomes.
• Negative EV doesn't mean you'll lose every time — it means you'll lose on average over many trials.
• For investments, EV is the expected return but variance determines risk — look at both.
• Consider the utility curve: losing $100 may hurt more than gaining $100 helps (loss aversion).
• Sensitivity analysis: change one probability at a time to see which outcome drives EV most.

Expected Value in Gambling and Casinos

Every casino game has a known expected value for the player. Blackjack with basic strategy: -0.5% EV. Roulette (double zero): -5.26% EV. Craps pass line: -1.41% EV. Slot machines: -2% to -15% EV. Professional advantage players find situations where EV is positive through card counting, matched betting, or promotional overlays.

Expected Value in Finance

Portfolio managers use expected return (a form of EV) combined with standard deviation to build efficient portfolios. The Sharpe ratio = (Expected Return - Risk-Free Rate) / Standard Deviation measures risk-adjusted performance. Higher Sharpe ratios indicate better risk-return trade-offs. Modern portfolio theory is built entirely on expected value and variance.

Decision Trees and Multi-Stage EV

Complex decisions with sequential outcomes use decision trees. Each branch has a probability and payoff, and you work backward from the endpoints to calculate the EV at each decision node. This approach handles problems like R&D investments (Phase 1 success → Phase 2 investment → market launch probability) where outcomes cascade.

Frequently Asked Questions

• What does expected value mean in simple terms?

  Expected value is the average result you'd get if you repeated an experiment many times. If a coin flip pays $10 for heads and $0 for tails, the EV is $5 — not a possible single outcome, but the long-run average.

• Why can expected value be misleading?

  EV ignores variance. A game with 50% chance of +$1M and 50% chance of -$999K has EV of +$500 but is extremely risky. The Kelly Criterion and expected utility theory address this by accounting for risk tolerance.

• How is expected value used in poker?

  Poker players calculate EV for every decision. If calling a $100 bet gives you a 30% chance to win a $400 pot, EV = 0.3 × $400 - 0.7 × $100 = +$50. Positive EV calls are profitable long-term.

• What if probabilities don't sum to 1?

  Probabilities must sum to exactly 1 (or 100%). If they don't, the distribution is incomplete. This calculator normalizes your inputs or warns you to fix them.

• Can expected value be negative?

  Yes. Most gambling games have negative EV for the player (the house edge). A fair roulette bet on red has EV = -$0.053 per $1 bet because of the green zero/double-zero.

• How is EV used in business decisions?

  Companies use EV to compare projects under uncertainty. If Project A has 60% chance of $1M profit and 40% chance of $200K loss, EV = $520K. Compare this to alternatives to pick the highest expected return adjusted for risk.