My rudimentary understanding of dice analysis consists of the following:
[li]The average roll for a given die is the top value plus one, divided by two. [/li][li]The probability that a combined event will occur is determined by multiplying the probability of each element not occurring, then subtracting that from 1.[/li][/ul]
Everything else I pretty much have to do by brute force generation of a histogram. But in games like Savage Worlds or HackMaster you get these exploding dice rules that I am powerless to analyze. Somewhere I got the data I needed to chart for Savage Worlds the results attainable vs. the die type used. But I’m now puzzling over HackMaster, for which I don’t see anybody having blazed the trail further than to have simulated die rolls. There, the results you get differ every time you attempt the analysis, and are subject to the vagaries that exist between random computer generation and the dice themselves.
With exploding dice systems, you need those kind of equations with N+1’s, brackets and ellipses. I don’t know how to do that. More importantly, I don’t even know how to brute force that with a spreadsheet. Well, okay, I can approximate the odds of results from one die. But I’m not sure how to even approximate the additive results of multiple results from one die type, much less multiple die types.
For HackMaster, there is a rule of Penetration. A roll of 1d4p means that you roll a 4-sided die, and if the result is the maximum 4, you roll again and add the next roll -1, and so on until you stop rolling the maximum value of the die (always only -1 per roll after the first).
From this, how is it possible to work out the likelihood of a given total result of multiple such die rolls added together?