Expected Value Calculator — Step-by-Step Solutions

The Expected Value Calculator computes E(X) from probability distributions, frequency tables, or game scenarios. It also calculates variance and standard deviation to give you a complete picture of the distribution.

Enter your values and their probabilities or frequencies, and get instant results with step-by-step working. The game mode tells you whether a game is fair or which side has the advantage.

Value (x)Probability

Your calculations will appear here

How to Use the Expected Value Calculator

Choose a mode: From Probability Table when you know the probability of each outcome, From Frequency Table when you have counts, or Expected Value of a Game for bets and payoffs.
Enter one row for each outcome: the value (x) in the first column and the probability or frequency in the second. Use Add row for more outcomes and the x button to delete one.
For probabilities, the values must add up to 1 (a small rounding tolerance is allowed). For a frequency table the counts are converted to probabilities for you.
In game mode, enter the net gain or loss for each outcome: a loss is a negative number. The result tells you the average win or loss per play and whether the game is fair.

Formula and Theory

The expected value is the long-run average of a random quantity: the value you would see on average if you repeated the experiment many times. For a discrete variable it is the sum of each outcome times its probability: E(X) = sum of x x P(x).

It does not have to be a possible outcome. A fair die has an expected value of 3.5, even though no roll can show 3.5. It describes the centre of the distribution, not a single result.

The variance, E(X^2) - (E(X))^2, and the standard deviation, its square root, measure how widely the outcomes spread around the expected value. Two bets can have the same expected value but very different risk.

In a game, the expected value of the net gain tells you whether the game favours you. A positive value means you win on average, a negative value means you lose on average, and zero is a fair game. Frequencies are turned into probabilities by dividing by the total count.

Worked Examples

Example 1: Fair die

Problem: Find the expected value, variance and standard deviation of a single roll of a fair die.

Solution: Each face has probability 1/6. E(X) = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 21 / 6 = 3.5. E(X^2) = (1 + 4 + 9 + 16 + 25 + 36) / 6 = 91 / 6 = 15.1667. Variance = 15.1667 - 3.5^2 = 15.1667 - 12.25 = 2.9167. SD = sqrt(2.9167) = 1.7078.

Answer: E(X) = 3.5, Var = 2.9167, SD = 1.7078

Example 2: From a frequency table

Problem: In a survey, 5 households have 0 pets, 12 have 1, 8 have 2 and 5 have 3. Find the expected number of pets per household.

Solution: Total households = 5 + 12 + 8 + 5 = 30. E(X) = (0 x 5 + 1 x 12 + 2 x 8 + 3 x 5) / 30 = (0 + 12 + 16 + 15) / 30 = 43 / 30 = 1.4333. E(X^2) = (0 + 12 + 32 + 45) / 30 = 2.9667, so variance = 2.9667 - 1.4333^2 = 0.9122 and SD = 0.9551.

Answer: E(X) = 1.4333 pets, SD = 0.9551

Example 3: A fair raffle

Problem: A ticket costs $10. There is a 10% chance of winning a $100 prize. Is the game fair?

Solution: Net outcomes: win $100 and pay $10 gives +$90 with probability 0.1; otherwise lose $10 with probability 0.9. E = 90 x 0.1 + (-10) x 0.9 = 9 - 9 = 0.

Answer: E = $0 per play, so the game is fair

Example 4: A game that favours the house

Problem: In American roulette (38 pockets), a $1 bet on one number pays $35 profit if it hits and loses the $1 otherwise. What is the expected result per spin?

Solution: P(win) = 1/38 and P(lose) = 37/38. E = 35 x (1/38) + (-1) x (37/38) = (35 - 37) / 38 = -2/38 = -0.0526.

Answer: About -$0.053 per $1 bet (the house keeps 5.26%)

Frequently Asked Questions

What is expected value in simple terms?

It is the average result you would get if you repeated a random process many times. It weights each possible outcome by how likely it is.

Can the expected value be a number that never happens?

Yes. A die has an expected value of 3.5 but never shows 3.5. The expected value describes the long-run average, not a single outcome.

Why must the probabilities add up to 1?

Because one of the listed outcomes must occur. If your probabilities sum to something else, the table is missing outcomes or has an error. The calculator allows only a tiny rounding gap.

How do I calculate the expected value of a bet?

List every outcome as the net gain or loss, with its probability, then add up value times probability. A negative total means you lose money on average.

What does a positive expected value guarantee?

Only an average over many plays. A single play can still lose, and with a high standard deviation the results can swing widely before the average shows up.

How is expected value related to the mean?

It is the mean of a probability distribution. For a data set where each observation is equally likely, the expected value equals the ordinary average.