Introduction to Probability

How to calculate expected value?

Quick answer

You calculate expected value by multiplying each possible outcome value by its probability of occurring, then adding all of these products together. The resulting number represents the long-term average payoff per play if the game or random experiment is repeated many times under identical conditions.

The topic explained

What expected value means

Expected value is the weighted average outcome of a random event when repeated many times. Calculating this figure helps you analyze whether a game, raffle, or financial decision is favorable over the long run.

It is important to remember that the expected value does not need to be an outcome that can actually happen in a single trial. For example, rolling a standard six-sided die has an expected value of 3.5, even though no roll will ever land on a half number.

The step-by-step calculation method

To find the expected value, first identify all possible numerical outcomes and their matching probabilities. Make sure the sum of all probabilities equals 1.

Next, multiply each outcome value by its probability. Finally, add all those products together. In mathematical notation, this is written as E(X) = Σ [x · P(x)], where x represents the value of an outcome and P(x) is its probability.

Evaluating fair games

A game is defined as fair when the expected net gain equals zero. This means that neither the player nor the host has an advantage over many rounds.

To find the entry price that makes a game fair, simply find the expected gross payout. For instance, if a carnival wheel pays $10 with a probability of 0.3 and $0 with a probability of 0.7, the expected payout is ($10 · 0.3) + ($0 · 0.7) = $3. Setting the ticket price at exactly $3 makes it a fair game.

Free printable worksheet

Calculating Expected Value and Fair Price Practice

Designed for 10th-grade probability students. Complete each problem by finding the expected payoff or net gain. Show your multiplication steps.

Print the worksheet Regenerate new worksheet

Free, no sign-up. The answer key prints on the second page. Regenerating a new worksheet on the same topic needs a free account.

Personal AI lessons and practice, from preschool to university

Free printable worksheetlessoons.com

Calculating Expected Value and Fair Price PracticeName:

A spinner has 4 equal sections labeled $2, $4, $8, and $10. What is the expected payout for one spin?
You roll a standard 6-sided die. Rolling a 6 pays $12, while any other number pays $0. Calculate the expected payout.
A bag contains 3 red marbles and 7 blue marbles. Drawing a red marble pays $30; drawing a blue marble pays $0. What is the expected payout?
A raffle sells 200 tickets at $5 each for one prize of $400. What is the expected net value of purchasing one ticket?
A game pays $20 with probability 0.1, $5 with probability 0.4, and $0 with probability 0.5. What entry fee makes this a fair game?
A game costs $2 to play. You flip two coins. If both are heads, you win $6; otherwise, you win $0. Is this game fair, favorable to the player, or unfavorable?

Common questions

Can expected value be a negative number?

Yes. When calculating net expected value, which takes the entry fee or initial bet into account, the result is frequently negative. A negative expected value indicates that a player will lose money on average over many trials.

Why is the expected value often not an actual possible outcome?

Expected value is a mathematical balance point, not a prediction of a single trial. It represents the mean of an infinite number of repetitions, which often results in a decimal value that no single round can produce.

What is the difference between gross expected value and net expected value?

Gross expected value accounts only for the prizes won. Net expected value subtracts the cost of entry from the gross winnings to show true average profit or loss per turn.

How does the Law of Large Numbers connect to expected value?

The Law of Large Numbers states that as the number of experiments grows, the experimental average gets closer to the theoretical expected value.

Still have a question? Ask Lessi

Any question about this subject. Lessi answers right here, step by step.

Try asking

Asking needs a free account; your question is kept while you sign up. Answers are written by AI and can contain mistakes. Lessi doesn’t give medical or legal advice.