Expected Value Calculator
Print PageAn Expected Value Calculator (also known as an E[X] Calculator, Probability Weighted Average Utility, Financial EV Calculator, or Decision Tree Risk Analyzer) calculates the long-run theoretical average outcome of a random variable X across discrete probability distributions (E[X] = ∑ xi · pi). Whether you are evaluating startup investment opportunities, analyzing insurance policy underwriting premiums, calculating casino house edge and poker pot odds, or assessing capital allocation decision trees, an expected value calculator determines whether a scenario yields a positive or negative long-run return.
In probability theory, Expected Value (E[X]) represents the center of mass of a probability distribution. If an experiment is repeated thousands of times, the empirical arithmetic mean of the outcomes converges to the expected value under the Law of Large Numbers.
Our free online Expected Value Calculator provides instant calculations across discrete outcomes, variance, standard deviation, and financial risk metrics:
- Discrete Expected Value Formula (E[X]):
E[X] = ∑i=1n xi · pi = x1p1 + x2p2 + ... + xnpn(where ∑ pi = 1.0). - Variance Formula (σ2):
Var(X) = E[X2] - (E[X])2 = ∑i=1n (xi - E[X])2 · pi. - Standard Deviation Formula (σ):
σ = √[ Var(X) ]. - Linearity of Expectation Rule:
E[aX + bY + c] = a·E[X] + b·E[Y] + c.
Master Business Venture Decision Tree Expected Value Table
The table below displays the probability distribution, individual weighted contributions, variance components, and final calculated expected value for a commercial product launch investment scenario:
| Market Scenario State | Financial Payoff (xi) | Probability (pi) | Weighted Product (xi · pi) | Squared Deviation (xi – E[X])2 · pi |
|---|---|---|---|---|
| High Market Success | +$500,000 | 0.20 (20%) | +$100,000 | $33,620,000,000 |
| Moderate Market Success | +$100,000 | 0.50 (50%) | +$50,000 | $50,000,000 |
| Product Launch Failure | -$200,000 | 0.30 (30%) | -$60,000 | $25,230,000,000 |
| TOTAL OVERALL EXPECTATION | N/A | 1.00 (100%) | E[X] = +$90,000 | Var = $58.9B &implies; σ = $242,693 |
Step-by-Step Casino Roulette & Venture Investment Example
To calculate the expected value of a $100 bet on a single number in American Roulette (38 total pockets: 1-36, 0, 00; win pays 35 to 1):
Step 1 (Identify Winning Payoff & Probability): Win Outcome x1 = +$3,500 with Probability p1 = 1 ÷ 38 ≈ 0.026316 (2.63%)
Step 2 (Identify Losing Payoff & Probability): Lose Outcome x2 = -$100 with Probability p2 = 37 ÷ 38 ≈ 0.973684 (97.37%)
Step 3 (Multiply Outcome by Probability): E[X] = (+$3,500 × 1/38) + (-$100 × 37/38)
Step 4 (Sum Weighted Contributions): E[X] = ($3,500 ÷ 38) - ($3,700 ÷ 38) = -$200 ÷ 38 = -$5.263158 ≈ -$5.26
Thus, for every $100 bet on a single roulette number, the player has an expected loss of -$5.26, representing an exact 5.26% casino house edge.
Positive Expected Value (+EV) vs. Negative Expected Value (-EV)
Below is a comparative reference chart detailing decision making criteria based on Expected Value signs:
| Expected Value Type | Mathematical Criterion | Long-Run Outcome Reality | Primary Real-World Examples |
|---|---|---|---|
| Positive Expected Value (+EV) | E[X] > 0 |
Generates long-run profit over many repetitions | Profitable business investments, advantage poker play, value betting |
| Negative Expected Value (-EV) | E[X] < 0 |
Guarantees long-run financial loss over time | Casino games (roulette, slots), lottery tickets, uncalculated risks |
| Fair Game (Zero EV) | E[X] = 0 |
Breakeven in the long run (zero house edge) | Flipping a fair coin with 1-to-1 payout |
History & Mathematics: 1657 Christiaan Huygens & Expectatio
1657 Christiaan Huygens & De Ratiociniis in Ludo Aleae
Formulated by Dutch polymath Christiaan Huygens in his 1657 treatise De Ratiociniis in Ludo Aleae (“On Reasoning in Games of Chance”), Huygens introduced the formal mathematical definition of Expected Value (expectatio). Huygens showed that in any game of chance, the fair price of a chance equals its mathematical expectation.
1654 Pascal-Fermat Correspondence
Three years earlier in 1654, French mathematicians Blaise Pascal and Pierre de Fermat laid the conceptual foundation by solving the Problem of Points, establishing that bets should be evaluated by weighting future outcomes by their probabilities.
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Frequently Asked Questions (FAQ)
What is Expected Value in simple terms?
Expected Value (E[X]) is the average result you would expect to get if you repeated a random experiment many times. It is calculated by multiplying each possible outcome by its probability and adding the results together.
Can Expected Value be negative?
Yes! A negative expected value (-EV) means that on average, you will lose money or points per trial over time (such as casino games or lottery tickets).
How do you calculate Expected Value for a business decision?
Multiply each potential profit or loss by its estimated probability of occurrence, then sum all values: E[X] = (Profit1 × P1) + (Profit2 × P2) + ... + (Lossn × Pn).