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Sasaisen
2015-02-06, 03:05 PM
Let me sketch out a scenario:

A character rolls 3d6 + (a bonus between 0 and 4) vs a DC of 10 after some interval of time has passed.
Every failed check imposes a -1 penalty to future checks.
After 10 failed checks in total, the character is dead.

What I want to know is the probability that a character will be dead after 10, 11, 12, etc. intervals. My statistics abilities are rusty, so is anyone able to lend a hand? I don't mind calculating it once I know how, I'm just not sure how to frame the problem mathematically.

Radar
2015-02-06, 03:45 PM
The basic thing you need to have is:
What is the probability of scoring 10+n or higher on a 3d6+bonus roll? Let's call this probability Pn. The easiest way to calculate that is to use a dice simulator (Anydice (http://anydice.com/) is great for it).

Once you have this, the rest is a pure death process (http://en.wikipedia.org/wiki/Birth%E2%80%93death_process).

You start with probability one of having no failed throws (state n=0 and the refered probability S0). In the next step, you put S0*P0 as S1 and S0*(1-P0) into S0. For each timestep and for each state (up to n=9, since you don't need the rest) you calculate Sn*Pn subtract it from Sn and add it to Sn+1. Best way is to do it from the highest state to the lowest, so you don't jump many steps in one time interval accidentialy.

When you want to know, what is the probability of being still alive after a given number of time steps, you just add up every state probability Sn from n=0 to n=9.

neonchameleon
2015-02-06, 04:04 PM
It's a finite banded solution with ten variables. The fast way to do it would be simply make an Excel table ten columns wide and with however many rows you need - each row being dependent on the one above.

Radar
2015-02-06, 05:12 PM
Errata

I should have written:
What is the probability of scoring 9+n or lower on a 3d6+bonus roll?

Bolding to emphasise the correction.

Sasaisen
2015-02-06, 07:32 PM
I had completely blanked on Markov chains. Thanks!