Determining dice probabilities

I have only managed to figure out how to do this with two dice by creating a result matrix and using countif() to find occurrences of a a particular sum.

I'm trying to create a formula, or whatever solution works, to determine what the probability of getting a certain sum on dice is.

Example: you roll 3 six sided dice and want to know the distribution of results for possibilities 3 through 18.

Also what if I want to change my dice to four six sided dice, but only keeping the highest three results? My range stays 3 to 18 but the mean and deviation should be higher and lower respectively.

Any thoughts?

Hi Ted,

See if the attached helps. In reality you have 6x6x6 possible outcomes to your die rolling problem. But the attached is more fun to deal with. Press F9 to see the results change.

3 Die Probability.xlsx

Not sure why you are interested in mean and [standard] deviation for this (normally you would just be interested in probabilities for the discrete values) but here is an exhaustive analysis that addresses both of the scenarios that you identified.

Pl see file. Columns C to G are for verification.

Formula for 3 dies. I2 is event.

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Formula for 4 dies. L2 is event.

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If you want exact answers, the probability of rolling a sum N on M 6-sided dice is the coefficient of x^N in

(x + x^2 + x^3 + x^4 + x^5 + x^6)^M / 6^M

You can expand polynomials like this at, for instance https://xrjunque.nom.es/polycalc.aspx

So for example,

(x+x^2+x^3+x^4+x^5+x^6)^4

returns

x^24 +4*x^23 +10*x^22 +20*x^21 +35*x^20 +56*x^19 +80*x^18 +104*x^17 +125*x^16 +140*x^15 +146*x^14 +140*x^13 +125*x^12 +104*x^11 +80*x^10 +56*x^9 +35*x^8 +20*x^7 +10*x^6 +4*x^5 +x^4

Which means the chance of rolling a 14 on three dice is 146/6^4 = 146/1296