Concepts → Fundamentals

Probability fundamentals, on dice

Five short pages, in order, that build up the language and the machinery the rest of /concepts assumes you have. Dice are the cleanest first example of probability — every outcome is countable, every probability is exact, and you can hold the whole distribution in your head. If you've never been formally introduced to probability distributions, expected value, variance, or convolutions, start here.

1. What's a probability distribution?
From 1d6 (flat) to 2d6 (a tent) to 3d6 (smoother still). Sample space, PMF, support, and the convolution that takes you from one to the next.
2. Expected value, in dice
The weighted average. Linearity of expectation — E[X + Y] = E[X] + E[Y], always — and the (M+1)/2 shortcut for fair 1dM dice. Why the mean is often the wrong question.
3. Variance and standard deviation
How spread-out the distribution is. The (M² − 1)/12 closed form for fair dice, additivity of variance under independence, and why SD(2d6) ≈ 2.41, not 2 · SD(1d6).
4. Independence, sums, and convolutions
What independence means, why summing dice is convolving PMFs, when the closed-form mean and variance shortcuts apply, and when they don't (advantage, reroll mechanics).
5. The normal approximation, and where it lies
8d6 is bell-shaped, but 1d20 is not, 1d6! is not, and 2d20kh1 is not. The central limit theorem, when to lean on it, and when to trust the engine instead.