About the Expected Value Calculator
Expected value distills an entire probability distribution down to one number, the long-run average outcome you'd expect if an uncertain event repeated many times. Our Expected Value Calculator finds it from a list of outcomes and their probabilities.
How It Works
The calculator multiplies each possible outcome by its probability, then sums all of those products together - a probability-weighted average across every possible result, after first checking that the entered probabilities actually sum to 1.
Formula & Methodology
Expected value works by weighting each possible outcome according to how likely it is, rather than treating every outcome as equally important the way a simple average would. An outcome with a high probability contributes more heavily to the total than one that's rare, even if the rare outcome has a large value - this weighting is exactly what makes expected value meaningful for decisions involving uncertainty, like whether a bet or investment is favorable on average over many repetitions.
Step-by-Step: Calculating It By Hand
- 1List every possible outcome and its associated probability.
- 2Confirm the probabilities sum to 1 (100%) - if they don't, something in the data is inconsistent.
- 3Multiply each outcome by its probability.
- 4Sum all of those products to find the expected value.
Examples
Simple bet
Outcomes of 10, −5, and 0 with probabilities 0.3, 0.5, and 0.2 give an expected value of (10×0.3) + (−5×0.5) + (0×0.2) = 0.5.
Die roll
A standard six-sided die, with each face equally likely at 1/6, has an expected value of (1+2+3+4+5+6)/6 = 3.5.
Advantages
- Validates that entered probabilities sum to 1 before calculating
- Shows each outcome's individual contribution to the total in a clear table
- Handles any number of discrete outcomes, not just a fixed few
- Useful for comparing the long-run fairness of different bets or decisions
Common Mistakes
- Entering probabilities that don't sum to 1, which invalidates the result
- Interpreting expected value as the most likely single outcome, rather than a long-run average
- Forgetting to include negative outcomes (losses) when evaluating a bet or decision
- Mismatching the order of outcomes and their corresponding probabilities
Edge Cases to Watch For
- If the probabilities don't sum to 1, the expected value calculation isn't meaningful, since some outcome is either missing or overcounted.
- Expected value can be a value the variable itself can never actually take - like 3.5 for a standard six-sided die roll, even though you can never roll 3.5.
- A negative outcome (like a loss in a bet) is entirely valid and pulls the expected value downward when weighted by its probability.
- An expected value near zero doesn't necessarily mean each individual outcome is small - large positive and negative outcomes can offset each other.
Common Use Cases
- Probability and statistics coursework
- Evaluating whether a game of chance or bet is favorable on average
- Risk analysis in finance and decision-making
- Insurance and actuarial-style calculations involving weighted outcomes