Dice & Probability | mysimulator.uk

One die: a uniform distribution

A single fair six-sided die is the simplest possible random variable: each face from 1 to 6 comes up with equal probability 1/6, giving a uniform discrete distribution. Its mean is the average of the six faces, (1+2+3+4+5+6)/6 = 3.5 — a value the die itself can never show, a useful reminder that an expected value is a long-run average, not a possible outcome. Its variance follows the standard formula for a discrete uniform distribution on {1,...,n}, (n²−1)/12, which for n=6 gives (36−1)/12 = 35/12 ≈ 2.9167, so the standard deviation of one die is about 1.708. Everything that follows — the shape of two-dice sums, the bell curve of ten dice, the odds a casino prices into a craps table — is built out of nothing more than this one flat, six-value distribution added to itself repeatedly.

Two dice: from uniform to triangular

The sum of two dice is not itself uniform, even though each individual die is. There are 36 equally likely ordered outcomes (6×6), but they collapse onto only 11 possible sums, from 2 to 12, and those sums are not equally represented: a sum of 7 can be made six different ways — 1-6, 2-5, 3-4, 4-3, 5-2, 6-1 — while a sum of 2 or 12 can only be made one way each (1-1 or 6-6). Plotting the count of ways to make each sum produces a perfect triangle, rising linearly from 1 at the extremes to 6 at the centre and back down, which is exactly what you get when you convolve two identical uniform distributions: convolution takes a spread and spreads it again, and the result is always smoother and more peaked in the middle than the inputs. This triangular shape is a preview — as more dice are added, the discrete convolution keeps softening the corners and pulling mass toward the centre, and by around n=10 dice the sum distribution already looks convincingly bell-shaped even though the source distribution, a single die, is about as far from a bell curve as a distribution can be.

counts = [1,1,1,1,1,1]         // one die: ways to roll 1..6

function convolve(a, b):
    result = array of zeros, length len(a)+len(b)-1
    for i in 0..len(a)-1:
        for j in 0..len(b)-1:
            result[i+j] += a[i] * b[j]
    return result

sums = [1]                     // generating function "1" (n=0 dice)
for k in 1..n:
    sums = convolve(sums, counts)

// sums[s] now holds the EXACT number of ways to roll total s
// e.g. for n=2: sums = [1,2,3,4,5,6,5,4,3,2,1]  (sums 2..12)

The Central Limit Theorem takes over

The formal reason the sum keeps looking more like a bell curve is the Central Limit Theorem: for n independent dice, the standardized sum (sum − nμ)/√(nσ²) converges in distribution to the standard normal distribution as n grows, regardless of the shape of the underlying distribution being summed — uniform, discrete and bounded though a single die is. Concretely, the sum of n dice has mean 3.5n and variance 35n/12, so its standard deviation is √(35n/12); for n=10 that is a mean of 35 with a standard deviation of about 5.4, and the actual discrete distribution of achievable sums, plotted as a histogram, is already visually indistinguishable from a Gaussian with those parameters. This is the same theorem that underlies why so many measurement errors, biological traits and aggregated sample means look approximately normal in practice — they are, in effect, sums of many small independent contributions, exactly like a sum of dice.

live demo · histogram of dice sums converging to a bell curve● LIVE

Law of Large Numbers, house edge and the maths behind the table

It is easy to conflate the Central Limit Theorem with the Law of Large Numbers, but they answer different questions. The Law of Large Numbers says that the average of n dice rolls — not the sum — converges to the true mean 3.5 as n grows; it is a statement about where the average ends up. The Central Limit Theorem says nothing about where; it describes the shape and rate of the fluctuations around that mean, telling you that those fluctuations are approximately Gaussian and shrink in relative size as 1/√n. A casino relies on both: the Law of Large Numbers guarantees that over enough rounds the house's average result converges to its theoretical edge, while the CLT-shaped variance around any individual session is what lets it size bankroll requirements and payout structures. The clearest application is a bet on the dice sum itself: a bet's expected value is payout × P(sum) − stake × P(not sum), and because 7 is the most probable two-dice sum (6/36, a bit under 1-in-6), craps games are built with the pass-line bet centred on 7 winning immediately on the first roll, with every other structure in the game — point numbers, odds bets — following from the same 36-outcome probability table.

Frequently asked questions

Why is 7 the most common sum of two dice?

There are 36 equally likely ordered outcomes when rolling two dice, and 7 can be made six different ways: 1-6, 2-5, 3-4, 4-3, 5-2 and 6-1. No other sum has that many combinations — 2 and 12 each have only one — so 7 sits at the peak of the resulting triangular distribution.

What's the actual difference between the Law of Large Numbers and the Central Limit Theorem?

The Law of Large Numbers says the average of many dice rolls converges to the true mean of 3.5 as the number of rolls grows — it tells you where the average ends up. The Central Limit Theorem says nothing about where; it describes the shape of the fluctuations around that mean, showing they become approximately Gaussian and shrink in relative size as the sample grows.

How do you compute the exact probability of a given sum for n dice without simulating?

Represent one die as the polynomial x + x^2 + ... + x^6, whose coefficients are the number of ways to roll each face. Raising that polynomial to the nth power — equivalently, convolving the six-element count array with itself n times — produces a new polynomial whose coefficients are the exact integer counts for every possible sum of n dice.

Try it live

Everything above runs in your browser — open Dice & Probability and add more dice to watch the histogram bend into a bell curve in real time. Nothing is installed, nothing is uploaded, the whole model lives in one tab.

▶ Open Dice & Probability simulation

Dice & Probability: The Central Limit Theorem at the Craps Table