Expected Value, Explained With Dice
The average of a thing that never happens, and why it is still the right number.
Expected value is the average outcome weighted by how likely each outcome is. It is the single most useful number in applied probability and the one most often misread, because it is an average of something that may never actually happen.
The expected value of a d6 is 3.5, and a d6 never rolls 3.5. The expected value of a lottery ticket is a fraction of its price, and no ticket has ever paid out a fraction of its price.
This page covers how to compute it, what it is good for, and the two situations where it is exactly the wrong thing to look at.
Computing it
Multiply each outcome by its probability and add the results. For a d6 that is (1+2+3+4+5+6) divided by 6, which is 3.5. For a coin flip paying 1 for heads and 0 for tails it is 0.5.
For unequal probabilities the same method applies with different weights. A draw paying 100 with probability 0.01 and 0 otherwise has an expected value of 1, regardless of how many times you draw.
The property that makes this useful is that expectations add. The expected total of 3d6 is 3 times 3.5, which is 10.5, and you do not have to enumerate the 216 combinations to know that. Expectations add even when the events are not independent, which is what makes the tool so widely applicable.
Expected values worth knowing
- 1d6: 3.5 · 2d6: 7 · 3d6: 10.5 · 1d20: 10.5
- A d20 and 3d6 have the same expected value and completely different distributions.
- That is the clearest possible demonstration that expected value is not a description of behaviour.
What it does not tell you
A d20 and 3d6 both have an expected value of 10.5, and they behave nothing alike. One gives every result a flat 5%; the other clusters two-thirds of its results between 8 and 13. Expected value cannot distinguish them.
That is the central limitation. Expected value collapses a whole distribution into one number, and everything about the shape — how spread out it is, whether it is symmetric, whether there is a small chance of a catastrophic result — is discarded.
The companion number is variance, or its square root, the standard deviation. For 1d20 the standard deviation is about 5.8; for 3d6 it is about 2.96. Those two numbers together tell you far more than either alone.
Why variance matters more in short games
Expected value is a long-run average, and the long run can be very long. In a game that lasts twenty rolls, the variance dominates completely and the expected value is nearly irrelevant to what actually happens.
A concrete case: two strategies with the same expected value but different variance are not equivalent in a short game. The high-variance strategy wins more of the games it wins by more, loses more of the ones it loses, and produces a wider spread of outcomes — which is better if you are behind and worse if you are ahead.
This is why competitive players talk about variance rather than expected value when the game is nearly over. A play with a lower expected value and a higher chance of the specific outcome you need is correct, and expected value alone would tell you otherwise.
- Expected value describes the long-run average, not any single game.
- In a 20-roll game, variance dominates and expected value barely constrains the result.
- Two options with equal expected value are not equivalent if their variance differs.
- When you need a specific outcome rather than a good average, variance is the number to optimise.
The case where it is dangerously wrong
Expected value assumes outcomes are additive and that you survive to keep playing. Where a bad outcome removes you from the game, it stops being the right measure entirely.
The standard illustration is a bet with a positive expected value and a small chance of losing everything. Repeated enough times, the ruinous outcome eventually occurs, and once it does no future positive-expectation bets are available. The expected value of each bet was genuinely positive and the strategy still loses.
This is why insurance is rational despite having a negative expected value for the buyer. Paying a premium reduces your expected wealth and removes an outcome you could not absorb, and the second consideration correctly outweighs the first.
Using it well
The honest summary is that expected value is the right starting point and rarely the right ending point. It tells you which option is better on average, over many repetitions, when you can absorb the bad cases.
Where those conditions hold — a long series of small decisions, no ruinous outcomes — it is close to the whole answer, and reasoning about anything else is usually overthinking.
Where they do not hold, ask two more questions: how wide is the spread, and is there an outcome I cannot come back from. Those two additions cover almost every case where a naive expected-value calculation goes wrong.
Frequently asked questions
What is the expected value of a d6?
3.5 — the average of 1 through 6. Note that a d6 can never actually roll 3.5, which is the point people find odd.
Why do a d20 and 3d6 have the same expected value?
Because both average 10.5. They behave completely differently, which shows that expected value discards everything about the shape of the distribution.
What should I look at alongside it?
Variance, or its square root the standard deviation. For 1d20 that is about 5.8; for 3d6 about 2.96.
Does expected value apply to a single roll?
Only loosely. It is a long-run average, and in a short game the variance dominates what actually happens.
When is expected value the wrong measure?
When a bad outcome removes you from the game. A positive-expectation bet with a small chance of ruin loses in the long run, because the ruin ends the sequence.
Is insurance irrational because its expected value is negative?
No. It reduces expected wealth and removes an outcome you could not absorb, and the second consideration correctly outweighs the first.
Do expectations add for dependent events?
Yes — that is what makes the tool so useful. The expected total of 3d6 is 3 × 3.5 without enumerating anything.
How many repetitions until the average shows up?
More than people expect. The gap between observed and expected shrinks with the square root of the count, so a hundred rolls is only ten times tighter than one.
Tools that use this
Related guides
Dice Probability Reference: d4 to d20
Single dice, sums, advantage, and dice pools — the numbers you actually look up.
The Law of Large Numbers, Practically
What 'it evens out' actually promises, on what timescale, and what it never promises.
How Weighted Random Selection Works
Cumulative sums, one float, and why duplicating entries is the wrong way to do it.