Why is the expected value something that cannot happen?

Because it is an average, not an outcome. A die averages 3.5 across many rolls and shows a whole number every time. Where no outcome sits near the average, the page says so.

Do the weights have to add up to 100?

No. They are relative and are normalised for you, so 1 and 9 means the same as 10% and 90%. Just do not mix percentages and relative weights in one list.

Is a positive expected value a good bet?

It means you would gain on average over enough repetitions. Whether that is a good idea depends on the spread, how many goes you get and what a bad run would cost you. This is arithmetic, not advice.

Expected Value Calculator

The expected value of a fair die is 3.5, which is a number the die can never show. That is not a curiosity — it is the whole point, and it is the thing people get wrong when they treat an average as a typical result. This works out the exact expected value and variance of any set of weighted outcomes, and tells you plainly when no single outcome is anywhere near the average.

Outcomes

One per line as “name, weight, value”. Weight is how likely relative to the others; value is what it is worth.

0.000

expected value — the long-run average per go. Standard deviation 3.000, so a typical result sits roughly that far either side of the average.

Each outcome, its probability and its contribution
Outcome	Chance	Value	Contributes
Win	10.00%	9	0.900
Lose	90.00%	-1	-0.900

No single outcome equals the expected value. That is normal — the expected value is a long-run average, not a result you can get.

Probabilities are worked out from your weights and always add to 100%. An expected value is what the average tends to over many goes — it says nothing about any single one, and it is not a prediction.

What this generator does

Computes the average and the spread together, which is what makes the answer honest. The dice and card probability calculators here give the chance of an outcome; this gives what the outcomes are worth on average and how far a typical result sits from it. Weights become probabilities automatically, so you can enter 1 and 9 rather than working out 10% and 90% first.

How to use this tool

List the outcomes, one per line, as “name, weight, value”.
Weight is how likely it is relative to the others; value is what it is worth.
Read the expected value, the spread, and the contribution of each outcome.

Understanding the controls

Outcomes: One per line as “name, weight, value”. Weights are relative and normalised to add up to 100%, so “1, 9” and “10, 90” describe the same thing. Values can be negative — a loss is a negative payoff.

Worked examples

A fair six-sided die: Expected value exactly 3.5; variance 35/12. No face shows 3.5.
10% chance of +9, 90% of −1: Expected value exactly 0 — a fair bet with a very unfair feel.
Weights of 1 and 9: Normalised to 10% and 90% automatically.
A single certain outcome: Expected value is that outcome; variance zero.

Common use cases

Working out whether a bet, an offer or a wager is worth taking on average
Comparing two options with different payoff shapes
Teaching the difference between the average and the typical
Checking a game mechanic's payout before committing to it
Putting a number on a decision where each branch has a value

How this generator works

Weights are divided by their total to give probabilities that sum to exactly one. The expected value is the sum of each probability times its value; the variance is the sum of each probability times its squared distance from that mean, and the standard deviation is its square root. Every figure is checked against the outcome table before it is shown, so the headline number always reconciles with the rows beneath it.

Randomness and fairness

Nothing here is random. It is exact arithmetic over the distribution you describe, and the same inputs always give the same answer.

For how randomness is produced across the whole site, see how Generate Random works.

Assumptions this tool makes

The listed outcomes are exhaustive and mutually exclusive — exactly one of them happens each time.

Limitations and good to know

An expected value describes the long run. It says nothing about one go, and a positive expected value can still lose money for a very long time.
It assumes the outcomes and their weights are right, which is usually the hard part.
Outcomes are independent draws from one set; sequences and conditional payoffs need the probability tree.
Nothing here is gambling advice, and a favourable expected value is not a reason to bet.

Common mistakes

Reading the expected value as what will happen: It is the average over many repetitions. The tool flags when no outcome is close to it, which is most of the time.
Ignoring the spread: Two options with the same expected value can be completely different propositions. The standard deviation is what tells them apart.
Entering probabilities that do not add to 100: You do not need to — enter relative weights and they are normalised. What you must not do is mix the two conventions in one list.

Practical tips

Enter losses as negative values rather than as a separate positive 'cost' outcome.
Compare two options by their expected value and their spread together, not by the average alone.

Privacy and your data

The outcomes you type stay in your browser. Nothing is uploaded, nothing is stored between visits, and none of it is written into the page address. Analytics records only that the tool was used.

Frequently asked questions

Why is the expected value something that cannot happen?: Because it is an average, not an outcome. A die averages 3.5 across many rolls and shows a whole number every time. Where no outcome sits near the average, the page says so.
Do the weights have to add up to 100?: No. They are relative and are normalised for you, so 1 and 9 means the same as 10% and 90%. Just do not mix percentages and relative weights in one list.
Is a positive expected value a good bet?: It means you would gain on average over enough repetitions. Whether that is a good idea depends on the spread, how many goes you get and what a bad run would cost you. This is arithmetic, not advice.