How can I take the convolution of two distributions on a spreadsheet such as Google Sheets?

Example 1: You want to multiply two arbitrary polynomials together on a spreadsheet symbolically (e.g. 2*x^4 + x times 3*x^2 - 5).

Example 2: Like in a board game or tabletop game, you have two 6-sided dice🎲🎲, with probabilities given by the following arrays/tables as a histogram. I'd like to figure what the probabilities are when you roll them together and add them up 🎲+🎲 "2d6".

x P(x) y P(y)
1 =1/6 = 16.6% 1 =1/6
2 =1/6 = 16.6% 2 =1/6
3 =1/6 = ... 3 =1/6
4 =1/6 4 =1/6
5 =1/6 5 =1/6
6 =1/6 6 =1/6

How do I get an array like below, in the general case with arrays of arbitrary size and values (not just a 6-sided die)?

sum P(X+Y=sum)
2 =1/36
3 =2/36
4 =3/36
5 =4/36
6 =5/36
7 =6/36
8 =5/36
9 =4/36
10 =3/36
11 =2/36
12 =1/36

(I would like to do this without using Google Apps Script, since I have found that sometimes excessive computation on Apps Script will hard-freeze the spreadsheet and cause it to become corrupted beyond repair for days or forever, with the "Loading..." error message.)

(disclosure: I am posting this question to answer it, which is explicitly allowed per the site rules, since the answer is nowhere on the internet.)

To convolve two or more distributions e.g. a * b * c, do the above operation one-at-a-time (e.g. conv(a, conv(b, c))) (ideally from the smallest domain to the largest domain will minimize computation; such an ordering will depend on your data, and is irrelevant if you don't have much data you're crunching).

2 Answers 2


This may be done as follows by using lambdas or Named Functions.

It is as simple as doing something like this:

=CONV(dice1, dice2)

This works as follows. The below code may be copy-pasted into a single cell, or the inner code (between 'begin function body' and 'end function body') may be copy-pasted into a named function's body. However, note that you will need the definition for MYMAP1 (see addendum at end of this answer far below) to workaround an existing bug in Google Sheets.

=LAMBDA(x_px, y_py,

named function: CONV (or whatever you want*)

description: given two Nx2 arrays {x,Px} and {y,Px} where x,y are values and Px,Py are probabilities, returns the convolution

parameter #1: x_px parameter #2: y_py

▼ ▼ ▼ begin named function body... ▼ ▼ ▼

LAMBDA(xs,pxs,ys,pys, flatouter2d, filterRows,

          flatouter2d(xs,ys, LAMBDA(a,b, a+b)),
          flatouter2d(pxs,pys, LAMBDA(a,b, a*b))
        "select Col1,sum(Col2) group by Col1"


...continue named function body... (the two functions below are the definitions of flatouter2d and filterRows used above)



        MAKEARRAY(len,1, LAMBDA(i,_,
          f(i, len-i, len)


▲ ▲ ▲ ...end named function body; ▲ ▲ ▲

below we apply it to the probability distribution of two 6-sided dice:

  {1,1/6; 2,1/6; 3,1/6; 4,1/6; 5,1/6; 6,1/6},
  {1,1/6; 2,1/6; 3,1/6; 4,1/6; 5,1/6; 6,1/6}

How it works:

  • The flatouter2d function creates a 2d table; for each row it considers the as (a's), and for that particular a, creates a row (transposes a column) by considering the bs (b's), and in considering the two writes the value f(a,b) into the cell (for some arbitrary function f; i.e. the dice values are summed f(a,b)=a+b, while the probabilities are multiplied f(a,b)=a*b. The flattening turns this X x Y array back into a single column.
  • We do this twice (once for the domain i.e. dice values, and once for the range i.e. probabilities), and paste them back side-by-side {..., ...} like a zipper to get back our original "{z,Pz}" (Nx2 array) format.
  • The duplicate entries are then summed together by using QUERY, combining the events with their associated probabilities.

(The query should probably have a sort by clause if one cares about sorting.)

The FILTERROWS function defined above is one of many possible versions and generically useful (equivalent to a more powerful "slice operator"); it is merely used to get rid of the first informational header row returned by QUERY i.e. i>0 filters out the header.

The function body for CONV could be less verbose (half the size) if FILTERROWS and even maybe FLAPMAP2D were moved into their own named functions.

*IMPORTANT NOTE: You may want to call this CONV_V1 or something, if you plan to have a name in Apps Script that is called conv, otherwise there will be a namespace collision and one or the other won't work.

If you are taking multiple convolutions, you can use the REDUCE function. For example, to take the power of a distribution DIST_POW(dist,n) e.g. sum of 4 six-sided dice:


bonus: To apply a function to the dice values DIST_MAP(dist, f):


e.g. like if 'snakeeyes' (1+1) is worth 12 then f=DIST_MAP(dist, LAMBDA(x, IF(x=2,12))`.


Bug in Google Sheets implementation of MAP:

It is currently the case that if you pass a 1x1 array to MAP, you will not be allowed to return a row or column. This can cause very frustrating and hard-to-track-down bugs. To avoid this problem, if the array you pass in might ever be of size 1, use one of the following workarounds:

definition of MYMAP1(xs, f):

=IF( (ROWS(xs)<>1)+(COLUMNS(xs)<>1),
  MAP(xs, f),

definition of MYMAP2(xs,ys, f):

=IF( (ROWS(xs)<>1)+(COLUMNS(xs)<>1) + (ROWS(ys)<>1)+(COLUMNS(ys)<>1)
  MAP(xs,ys, f),

You will need to add these definitions to your Named Functions.

(Also note that technically map({}, ...) should return the empty array {}, but I'm not sure that's even possible in Google Sheets to have an empty array.)

  • Thank you for contributing that. Regarding map(): I agree that the "no zero-length array allowed" issue is a problem. My understanding is that map() will always return an array of the same size as its arguments, including an array of just one value, as in =map( { 1 }, lambda(value, value + 1) ). You can resize an array of one value, one row or one column to match the dimensions of a range with arrayformula(iferror(range/0, value)). Dec 3, 2022 at 11:51

Not attempting to solve convolutions in the general case — just pointing out that you can get the probability distribution of the sum of two simple independent integer variables with this pattern:

    dice1, dice2, 
        flatten(dice1 + transpose(dice2)), 
        "select Col1, count(Col1) / " & numCombos & " 
         group by Col1 
           Col1 'sum', 
           count(Col1) / " & numCombos & " 'P(X+Y=sum)' 
      counta(dice1) * counta(dice2) 
    sequence(6), sequence(6) 

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