Question 1

What is expected value?

Accepted Answer

The average result you would get per try over many tries: the sum of each outcome times its probability, μ = Σ x · P(x). A game that wins $10 half the time and loses $5 half the time has an expected value of $2.50.

Question 2

How do I enter the outcomes?

Accepted Answer

One outcome per line: the value, then its probability as a percent or a decimal — "10 50%" or "10 0.5".

Question 3

What if the probabilities do not add up to 100%?

Accepted Answer

Then some outcomes are missing or the numbers are off — the calculator warns you. Probabilities of all possible outcomes must add up to exactly 100%. A common slip is to leave out the outcome where nothing happens, such as the losing ticket.

Question 4

What does the standard deviation tell me?

Accepted Answer

How far results typically land from the expected value — $7.50 in the game above, so single plays swing widely even though the average is positive. OpenStax’s formula is σ = √[Σ (x − μ)² · P(x)].

Question 5

Worked example: a digit lottery

Accepted Answer

OpenStax Example 4.5 describes a game where you pay $2 to pick five digits from 0 to 9, and win $100,000 if all five match in order. The chance is 1 in 100,000, or 0.001%. Enter the two outcomes as “100000 0.001%” and “−2 99.999%”. The expected value is 100,000 × 0.00001 + (−2) × 0.99999 = −0.99998, about −$1: over many plays you lose a dollar per game on average. The standard deviation is $316, far larger than the average, because almost every play loses $2 and a rare one pays a fortune. Example 4.6 is a biased coin that lands heads two thirds of the time: tails wins $10 and heads loses $6, so the expected value is 10 × 1/3 − 6 × 2/3 = −0.67 per play.

Question 6

Is a positive expected value a sure win?

Accepted Answer

No — it is a long-run average. Any single try can lose. OpenStax ties this to the Law of Large Numbers: as the number of trials grows, the average result moves toward the expected value, but over a handful of trials chance dominates.

Question 7

Can the expected value be a result that never happens?

Accepted Answer

Yes. A fair die shows 1 to 6 with equal chances, so its expected value is 3.5, a number no roll can produce. The expected value is a balance point, not a prediction for one trial.

Question 8

Common mistakes with expected value

Accepted Answer

First, entering the prize instead of the net result. If a ticket costs $2 and pays $10, the winning outcome is +$8 and the losing outcome is −$2; entering 10 and 0 overstates the value by the price of the ticket. Second, forgetting the sign of the losing outcome. Third, using probabilities that do not sum to 100%; the calculator flags this, but an outcome left out silently removes its share. Fourth, treating the expected value as the typical result when the distribution is lopsided, as in the lottery, where the usual outcome is the loss of $2. Fifth, writing a bare 5 for 5%: without the percent sign the number is read as a decimal, so 5 would mean 500%. Write 5% or 0.05.

Expected value calculator