Expected Value Calculator
Free expected value calculator. E(X) = Σ x·P(x) for discrete random variables. Variance, SD, probabi
Why This Statistical Analysis Matters
Why: Statistical calculator for analysis.
How: Enter inputs and compute results.
E(X) = Σ x·P(x) — Discrete Random Variables
Compute expected value, variance, and standard deviation for any discrete random variable. Casinos, lotteries, insurance, stocks, and decision-making.
Real-World Scenarios — Click to Load
Outcomes (1–20 rows)
Outcomes by Expected Contribution (x × P(x))
| Outcome | Value (x) | P(x) | x × P(x) |
|---|---|---|---|
| Outcome 3 | 2 | 0.3400 | 0.6800 |
| Outcome 2 | 1 | 0.3300 | 0.3300 |
| Outcome 1 | 0 | 0.3300 | 0.0000 |
Probability Distribution
Expected Contribution (x × P(x))
Probability Distribution (Doughnut)
Calculation Breakdown
⚠️For educational and informational purposes only. Verify with a qualified professional.
Key Takeaways
- • Expected value E(X) is the long-run average outcome — you may never actually get this exact value in a single trial
- • If E(X) is negative for a bet, you lose money on average — this is how casinos, lotteries, and insurance companies profit
- • Probabilities MUST sum to 1 for the expected value to be meaningful
- • E(aX + b) = a × E(X) + b — expected value is linear
- • Variance measures how spread out the outcomes are around E(X)
Did You Know?
How It Works
1. Define Outcomes and Probabilities
List each possible outcome x_i and its probability P(x_i). Probabilities must sum to 1.
2. The Summation: E(X) = Σ x_i P(x_i)
Multiply each outcome by its probability and sum. This gives the long-run average.
3. Variance: Measuring the Spread
Var(X) = E(X²) - [E(X)]². Higher variance means more uncertainty.
4. Decision Making
When choosing between options, pick the one with the highest E(X) — unless you care about variance (risk).
Expert Tips
Negative EV = Bad Bet
If E(X) < 0 for a gamble, don't take it (unless you value the entertainment)
EV doesn't mean you'll get that
E(X) = 3.5 for a die but you'll never roll 3.5
Consider variance too
Two bets with same E(X) but different variance have different risk profiles
Use for decision-making
Choose the option with highest E(X) when comparing alternatives
Why Use This Calculator vs Other Tools?
| Feature | This Calculator | EMV Calculator | Excel | R/Python |
|---|---|---|---|---|
| General discrete RV | ✅ | ⚠️ Finance only | ✅ | ✅ |
| Custom outcomes | ✅ | ⚠️ Limited | ✅ | ✅ |
| Variance & SD | ✅ | ⚠️ | ⚠️ Manual | ✅ |
| Probability charts | ✅ | ⚠️ | ❌ | ⚠️ |
| Contribution breakdown | ✅ | ⚠️ | ❌ | ⚠️ |
| Educational content | ✅ | ❌ | ❌ | ❌ |
Frequently Asked Questions
What is expected value in simple terms?
Expected value is the long-run average outcome. If you repeated the experiment many times, E(X) is what you would average. For a fair die, E(X) = 3.5 — even though you can never roll 3.5.
Can expected value be negative?
Yes. A negative E(X) means you lose on average. Casino games, lotteries, and most gambling have negative expected value for the player.
Why do casinos always win in the long run?
Every game has a built-in house edge — negative expected value for the player. Over millions of bets, the law of large numbers ensures the casino profits.
What happens if probabilities don't sum to 1?
The expected value becomes meaningless. E(X) assumes a valid probability distribution. Normalize your probabilities or fix the inputs.
How is expected value used in decision making?
When choosing between options with uncertain outcomes, pick the one with the highest E(X). But also consider variance — higher variance means more risk.
What is the difference between expected value and most likely outcome?
Expected value is the weighted average. Most likely outcome is the one with the highest probability. They can differ — e.g., a lottery has high E(win) from a tiny jackpot chance but the most likely outcome is losing.
What is the St. Petersburg Paradox?
A coin-flip game: flip until heads, win 2^n dollars. The expected value is infinite, yet no one would pay much to play. It led to utility theory — people value certainty.
How does expected value relate to the law of large numbers?
The law of large numbers says the sample mean converges to E(X) as the number of trials increases. That is why casinos always win — they run enough trials for the average to stabilize.
Expected Value by the Numbers
Official Data Sources
Disclaimer: This calculator provides expected value for discrete random variables for educational and decision-making reference. Probabilities must sum to 1 for valid interpretation. For continuous distributions, use integration.
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