Expected value is one of the most useful ideas in probability, and dice are the friendliest way to learn it. This quiz generates a fresh question every time you press the button, choosing the dice at random from d4, d6, d8, d10, d12 and d20 and two to five dice, so you can practise until the formulas feel natural.
Questions cover six skills: the average of a single die, the average total of several dice, adding a flat bonus to a damage roll, the standard deviation of a total, the average result of rolling with advantage, and the most likely total of a sum of dice. Each question has four answer options; the wrong ones are typical mistakes, such as dividing the number of sides by two, forgetting that a bonus is added once, or multiplying the standard deviation instead of the variance.
Pick a question type or leave it on the mix, choose your answer to see whether you were right, and read the worked solution underneath. If you get stuck, the dice average calculator on this site shows the full distribution for any combination of dice.
How to use the Dice Average Quiz
- Question type
- Your answer
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
Each question is built from random dice; the correct answer comes from the exact formulas and the distractors from common errors. Changing your answer keeps the same question; pressing “New question” (or changing the question type) generates a new one.
Worked example
Question: “You roll 2d8 and keep the higher result. What is the average kept value?” The chance that the kept value is k is (2k − 1)/64. Summing k × (2k − 1)/64 for k = 1 … 8 gives (8 + 1)(4 × 8 − 1)/(6 × 8) = 279/48 ≈ 5.813, compared with 4.5 for a single d8.
Formulas to remember
| Quantity | Formula | Example (d6) |
|---|---|---|
| Average of one die | (s + 1)/2 | 3.5 |
| Variance of one die | (s² − 1)/12 | 2.917 |
| Average of n dice | n(s + 1)/2 | 3d6 → 10.5 |
| SD of n dice | √(n(s² − 1)/12) | 3d6 → 2.958 |
| Average with advantage | (s + 1)(4s − 1)/(6s) | 4.472 |
Formula
E(one die) = (s + 1)/2 · E(n dice) = n(s + 1)/2 · SD(n dice) = √(n(s² − 1)/12) · E(higher of two dice) = (s + 1)(4s − 1)/(6s)
Worked example
Inputs — Question type: Mixed – any type; Your answer: Show me the solution.
Results — Question = You roll 2d12 and keep the higher result (advantage). What is the average kept value?; Choices = A) 6.5
B) 8.486
C) 13
D) 9.75; Worked solution = Answer B: 8.486. P(kept = k) = (2k − 1) / 144. Summing k × (2k − 1) / 144 for k = 1…12 gives (12 + 1)(4 × 12 − 1) / (6 × 12) = 8.486, about 1.986 more than a single roll..
Key terms
- Expected value
- The average outcome over many repetitions.
- Distractor
- A plausible wrong answer in a multiple-choice question.
- Advantage
- Rolling two dice and keeping the higher result.
Frequently asked questions
What is the average roll of a d20?
10.5 – the mean of the numbers 1 to 20, (1 + 20)/2.
Why is the average of a die not a whole number?
The expected value is a long-run average, not a face you can roll. For an even number of sides it falls halfway between the two middle faces.
How do I find the average of several dice?
Add the averages: 4d6 averages 4 × 3.5 = 14, and 2d8 + 1d6 averages 4.5 + 4.5 + 3.5 = 12.5.
How do I add a bonus?
Add it once to the dice average: 2d6 + 3 averages 7 + 3 = 10.
Why do I add variances rather than standard deviations?
For independent dice the variances add. The standard deviation of 4d6 is √(4 × 35/12) ≈ 3.42, not 4 × 1.71.
What is the average of a d20 rolled with advantage?
13.825 – about 3.3 more than a normal roll.
How many questions are there?
The questions are generated at random, so there are hundreds of combinations of dice and question types.