Expected Value Calculator
Calculate the expected value of a random variable from outcomes and their probabilities. Enter values for instant results with step-by-step formulas.
Formula
E(X) = sum of (x_i x P(x_i)) for all outcomes i
The expected value E(X) is calculated by multiplying each possible outcome value x_i by its probability P(x_i) and summing all products. Variance is calculated as E(X^2) - [E(X)]^2, and standard deviation is the square root of variance.
Worked Examples
Example 1: Investment Decision Analysis
Problem: An investment has three possible outcomes: $100 profit (30% chance), $50 profit (50% chance), or $80 loss (20% chance). Calculate the expected value.
Solution: E(X) = ($100 x 0.30) + ($50 x 0.50) + (-$80 x 0.20)\nE(X) = $30.00 + $25.00 + (-$16.00)\nE(X) = $39.00\nVariance = (100^2 x 0.30 + 50^2 x 0.50 + 80^2 x 0.20) - 39^2\nVariance = (3000 + 1250 + 1280) - 1521 = 4009\nStd Dev = sqrt(4009) = 63.32
Result: Expected Value: $39.00 | Std Dev: $63.32 | Favorable (positive EV)
Example 2: Lottery Ticket Expected Value
Problem: A $2 lottery ticket has a 1/1000 chance of winning $1000, 1/100 chance of winning $20, and 989/1000 chance of winning $0.
Solution: E(X) = ($1000 x 0.001) + ($20 x 0.01) + ($0 x 0.989) - $2 ticket cost\nE(X) = $1.00 + $0.20 + $0.00 - $2.00\nE(X) = -$0.80\nFor every $2 ticket, you lose $0.80 on average\nReturn on investment: -40%
Result: Expected Value: -$0.80 per ticket | Unfavorable (negative EV)
Frequently Asked Questions
What is expected value and why is it important in probability?
Expected value is the long-run average outcome of a random variable when an experiment is repeated many times. It is calculated by multiplying each possible outcome by its probability and summing all the products. The concept is fundamental in probability theory, statistics, economics, and decision-making because it provides a single number that summarizes the central tendency of a probability distribution. For example, if a game pays $10 with 50% probability and $0 with 50% probability, the expected value is $5. This does not mean you will ever win exactly $5, but over thousands of plays, your average winnings will converge to $5 per play. Insurance companies, casinos, and investors all rely heavily on expected value calculations.
How do you calculate expected value step by step?
To calculate expected value, follow these steps. First, list all possible outcomes of the random event or experiment. Second, assign a probability to each outcome, ensuring all probabilities sum to exactly 1.0 or 100 percent. Third, multiply each outcome value by its probability to get the weighted value. Fourth, sum all the weighted values to obtain the expected value. For example, consider a dice game where rolling 1-2 wins $30, rolling 3-4 wins $10, and rolling 5-6 loses $20. The calculation is E(X) = $30 x (2/6) + $10 x (2/6) + (-$20) x (2/6) = $10.00 + $3.33 + (-$6.67) = $6.67. The positive expected value indicates this game favors the player over many repetitions.
What is the difference between expected value and variance in probability?
Expected value and variance describe different aspects of a probability distribution. Expected value measures the central tendency or average outcome, telling you where the distribution is centered. Variance measures the spread or dispersion of outcomes around the expected value, indicating how much variability or risk is involved. Two distributions can have identical expected values but very different variances. For instance, receiving $100 with certainty has an expected value of $100 and zero variance. A coin flip paying $200 or $0 also has an expected value of $100 but high variance of $10,000. The standard deviation, which is the square root of variance, is often more interpretable because it is expressed in the same units as the outcomes.
Can expected value be negative and what does that mean?
Yes, expected value can be negative, and this indicates that on average, the outcome results in a loss rather than a gain over many repetitions. Most casino games have negative expected values for the player, which is how casinos maintain profitability. For example, American roulette has an expected value of approximately negative 5.26 cents per dollar wagered because of the house edge created by the zero and double-zero pockets. A negative expected value does not mean you will always lose, as short-term results can vary widely. However, the law of large numbers guarantees that over many repetitions, your average result will converge to the expected value. Understanding negative expected value helps rational decision-makers avoid systematically unfavorable bets.
How is expected value used in real-world decision making?
Expected value is widely applied across numerous fields for rational decision-making under uncertainty. In finance, investors use expected returns weighted by probability scenarios to evaluate portfolios and compare investment options with different risk profiles. Insurance companies calculate expected claim costs to set premium prices that ensure long-term profitability while covering policyholder losses. In healthcare, expected value analysis helps evaluate treatment options by weighting health outcomes by their likelihood of occurrence. Project managers use expected monetary value to quantify risks and determine appropriate contingency budgets. Game theory applications include poker strategy, where players calculate expected value of each possible action to determine optimal play. Even everyday decisions like choosing insurance deductibles or warranty purchases benefit from expected value thinking.
What is the law of large numbers and how does it relate to expected value?
The law of large numbers is a fundamental theorem in probability that states the average of results obtained from a large number of trials converges to the expected value as the number of trials increases. This means that while individual outcomes can deviate wildly from the expected value, the average over many repetitions will approach the mathematical expectation. For example, a fair coin has an expected value of 0.5 heads per flip, but you might flip 7 heads in 10 tries. Over 10,000 flips, however, the proportion of heads will be very close to 50%. This principle underpins casino profitability, insurance pricing, and portfolio diversification strategies.