[RPG] Help Needed With Probability Math for 2d10


In my Old School D&D campaign, I instituted a house rule to help the PCs raise their ability scores. Each ability score has a "fractional ability score". This is a percentile rating that starts at 0% at first level. Every time a PC levels up, they roll 2d10 for each fractional ability score and add it to the total.

Bob the second level fighter has a Strength score of 16, with a fractional strength of 09. When he levels up, he rolls 2d10 for his fractional strength and gets 12. His fractional strength is now 21.

When a fractional ability score gets to be 100, the ability score is raised by a point. Any fractional ability score points over 100 are retained, so if you had 108 points in Charisma when you leveled up, you'll gain a point of Charisma and retain the 08 fractional points.

My question is, how many times, on average, will you need to level up to gain an ability score increase? I know the worse case scenario takes 50 levels, and the best case scenario takes 5 levels, but what is the average?

(Related to this question: Rules for increasing ability scores in Basic D&D?)

(I asked this question on SE.Mathematics here: https://math.stackexchange.com/questions/733942/help-with-the-probabilty-of-rolling-two-ten-sided-dice-multiple-times-until-100 and am voting to close this question as off-topic.)

Best Answer

The mean result of 2d10 is 11. A reasonable approximation for number of levels between increments is therefore 9 (because 99 is closest, and you cannot have fractional levels).

The probability of having an increment by level 9 is slightly less than 50%. The probability of having the increment by level 10 is somewhat greater than 50%.

My calculations and tests show 9.6 levels mean before a stat increment under this system. I will spare the details - some discussion in meta about how complex Q&A here should be for probabilities before you are better off asking at https://math.stackexchange.com/

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