Adding Weighted Dice with a Visual Convolution

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This 3Blue1Brown short shows how to calculate the sum of two independent weighted dice by multiplying outcome probabilities and adding along diagonals: a discrete convolution.

1GridA grid lists all outcome pairs.
2Pair probabilityFor independent dice, multiply face probabilities.
3Equal sumsEqual sums lie on diagonals.
4Add diagonalsAdd pair probabilities on each diagonal.
5ConvolutionThe resulting distribution is a convolution.

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A grid lists all outcome pairs.
For independent dice, multiply face probabilities.
Equal sums lie on diagonals.
Add pair probabilities on each diagonal.
The resulting distribution is a convolution.

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Weighted dice

List and weight outcome pairs.

Suppose I roll two weighted dice and want the probability of each possible sum. This short illustrates the computation for independent rolls; the full video linked in its description extends the idea to continuous distributions.

I start with a six-by-six grid whose cells represent all pairs of outcomes. The dice need not be fair, so each face can have its own probability.

For independent dice, the probability of a pair is the probability of the first face multiplied by the probability of the second. If the rolls are dependent, a joint distribution is needed instead.

Diagonal sums

Combine pairs with equal totals.

The probabilities can be shown as heights in a three-dimensional plot. Pairs with the same sum lie on the same diagonal of the grid.

For example, one diagonal collects outcomes summing to six and another collects outcomes summing to seven. Adding the heights along each diagonal gives the probability of that sum.

Collapsing the plot along the diagonals produces the full distribution of the sum. This operation is discrete convolution of the two independent dice distributions.