Concepts → Fundamentals → What's a probability distribution?

What's a probability distribution? `1d6` vs `2d6`, and why one is flat and the other is a tent.

A probability distribution is a list of every outcome an experiment can produce, paired with the chance of each one happening. Dice are the cleanest first example: every outcome is countable, every probability is exact, and you can hold the entire distribution in your head at once.

One die: a uniform distribution

Roll a fair six-sided die. There are six outcomes — 1, 2, 3, 4, 5, 6 — and each is equally likely. We write that as P(X = k) = 1/6 for every k from 1 through 6. Every bar in the chart below has the same height because every face is equally likely. That's what uniform means.

1d6

min 1 max 6

1 16.67%
2 16.67%
3 16.67%
4 16.67%
5 16.67%
6 16.67%

The chart is the distribution: the height of the bar at outcome k is P(X = k). A few useful things you can already read off:

Every probability sits between 0 and 1 (we'd be in trouble otherwise — that's the definition of a probability).
The probabilities sum to 1: 6 × 1/6 = 1. Some outcome must happen, so the column heights have to add up.
Range: the smallest outcome is 1, the largest is 6. There is zero probability anywhere outside that range.

This shape — flat across the support — is called a discrete uniform distribution. It's the simplest non-trivial example of probability, and every other distribution on this site is built up from it.

Two dice: a tent shape

Now roll two six-sided dice and add the faces. The outcomes go from 2 (rolling 1+1) to 12 (rolling 6+6) — but they are not equally likely.

2d6

min 2 max 12

2 2.78%
3 5.56%
4 8.33%
5 11.11%
6 13.89%
7 16.67%
8 13.89%
9 11.11%
10 8.33%
11 5.56%
12 2.78%

The chart isn't flat. The middle outcome — 7 — is the most likely; the extremes are the least likely. Why?

There are 36 equally-likely (die₁, die₂) pairs. Six of them sum to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). Only one sums to 2: (1,1). Only one sums to 12: (6,6). The probability of each total is the count of pairs that produce it, divided by 36. So P(X = 7) = 6/36, P(X = 2) = 1/36. Same exact rationals you see in the bars above — tap any of them on the live chart to read the fraction back.

That triangular ramp-up-and-down is the visual fingerprint of a sum of two uniform distributions. It is not the bell-shaped curve of the normal distribution — yet. Add more dice and it gets smoother (see the normal approximation page); the tent is what convolution looks like after one step.

Three dice: smoother still

The same construction with three dice. Sum range 3 to 18; the distribution rises from 1/216 at the extremes, peaks around 10–11, and falls symmetrically on both sides.

3d6

min 3 max 18

3 0.46%
4 1.39%
5 2.78%
6 4.63%
7 6.94%
8 9.72%
9 11.57%
10 12.50%
11 12.50%
12 11.57%
13 9.72%
14 6.94%
15 4.63%
16 2.78%
17 1.39%
18 0.46%

Each step from 1d6 to 2d6 to 3d6 is a convolution — the operation that gives you the distribution of X + Y when you know the distributions of X and Y independently. That is what the engine on this site does, billions of times a day, in exact rationals.

Vocabulary you now have

Sample space — the set of possible outcomes. For 1d6 it's the integers 1 through 6; for 2d6 it's 2 through 12.
Probability mass function (PMF) — the function that maps each outcome to its probability. The bar chart is literally the PMF.
Support — the outcomes with non-zero probability. For a fair 1d6 the support is 1 through 6, and the PMF is zero everywhere else.
Discrete uniform — a PMF that's constant on its support. 1d6 is the canonical example.
Convolution — the operation that produces the distribution of a sum of independent random variables. Sums of dice are convolutions.

Try it yourself

↦ /strike/1d6 — uniform, every probability is 1/6 ↦ /strike/2d6 — the tent ↦ /strike/3d6 — already smoothing toward a bell ↦ /strike/4d6 — the bell is unmistakable

Tap any bar in any of those charts and the page shows you the exact rational probability of that outcome — not a rounded decimal, the actual fraction the engine computed.