Turning a matrix matrix into a probability table

Question

I want to build a probability table that is able to answer questions such as "if i roll 5, 4 sided dice what is the probability i get at least a 16?"

My current code is:

numberOfdice = 5;
diceSize = 4;
sampleSpace = Tuples[Table[Range[1, diceSize], {numberOfdice}]];
list = Sort[Total[sampleSpace, {2}]];
min = numberOfdice;
max = numberOfdice*diceSize + 1;
counts = BinCounts[list, {min, max, 1}];
Transpose[{Range[min, max - 1], counts}] // MatrixForm

Which gives me a matrix where the left hand column is the sum of the dice, and the right hand column is the count of that total in the sample space.

But im not sure what to do from here, does anyone have any advice?

Answer 1

Generating tuples and then machinating over them will get terribly inefficient very quickly (for example, try 10D20 with your code: you'll blow memory before getting anywhere).

Much easier, and efficient, to appeal directly to a probabilistic result.

Here's a function that given the number of dice and number of faces produces a grid with sums and various probabilities.

probgrid[numberOfdice_, diceSize_] := 
  Module[{d, face, probs, aprobs},
   probs = 
    CoefficientList[Sum[d^face, {face, diceSize}]^numberOfdice, 
       d][[numberOfdice + 1 ;;]]/diceSize^numberOfdice;
   aprobs = Accumulate@probs;
   
   Grid[Prepend[
     Transpose[{Range[numberOfdice, numberOfdice diceSize], probs, 
       aprobs, 1 - aprobs + probs, Most[Prepend[aprobs, 0]], 
       1 - aprobs}], {"Sum", "=", "<=", ">=", "<", ">"}], 
    Frame -> All]];

Your example:

probgrid[5, 4]

If you want a decimal result instead of exact, just apply N:

N@probgrid[5, 4]