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.

Example: 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.)

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    \$\begingroup\$ This question appears to be off-topic because it is a mathematical question about probabilities and rate of change, and only incidentally involves RPGs \$\endgroup\$ – doppelgreener Mar 31 '14 at 13:39
  • \$\begingroup\$ @Discord Our rule of thumb on real-world topics (including math) lies in our What topics can I ask about here? help, right after the bullet points under the "This is not the right site for questions about" heading, and it doesn't really pass. There's nothing drawing from RPG.SE expertise here so it probably is more appropriate on a mathematics site. In fact, you'll probably get a better and more specific answer from a mathematician, who could tell you everything important you need to know about calculating this stuff. \$\endgroup\$ – doppelgreener Mar 31 '14 at 14:03
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    \$\begingroup\$ I don't understand why this question is being voted to close when there are many similar questions under the Probability tag that are just fine, eg rpg.stackexchange.com/questions/14690/… rpg.stackexchange.com/questions/34173/… \$\endgroup\$ – Wibbs Mar 31 '14 at 14:06
  • \$\begingroup\$ @Phil Previous similar questions being open is rarely an indicator new questions should stay open. Each should be judged on their own merits, and if anything it calls into question whether the older ones should remain open (our standards change over time). The first, though, is open because it is actually relevant to RPGs: it's asking about to-hit odds of a particular roll, and RPG experts can provide a better, different, or more specific answer compared to a mathematician. About the same is the case for #2. Not the case here. \$\endgroup\$ – doppelgreener Mar 31 '14 at 14:08
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    \$\begingroup\$ See related meta question for further discussion meta.rpg.stackexchange.com/questions/3401/… \$\endgroup\$ – Wibbs Mar 31 '14 at 14:15

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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    \$\begingroup\$ A Monte-Carlo simulation is not needed here. You can prove this trivially. \$\endgroup\$ – Sardathrion - against SE abuse Mar 31 '14 at 14:04
  • \$\begingroup\$ @Sardathrion: Yes ok, it was just as easy, and agrees with the result. \$\endgroup\$ – Neil Slater Mar 31 '14 at 15:25
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    \$\begingroup\$ You're missing the fact that he's adding 2d10 to each ability score. Each one has an average of 9 levels, but you're likely to get more than one in those 9 levels. \$\endgroup\$ – C. Ross Mar 31 '14 at 16:50
  • \$\begingroup\$ Also, for stats you want stats.stackexchange.com, not math. \$\endgroup\$ – C. Ross Mar 31 '14 at 16:51
  • \$\begingroup\$ @C.Ross: Yes I have not covered distribution amongst multiple abilities. I'll wait for now, as it adds another layer of complexity and the question is on hold. Probability questions AFAIK are OK in both stats and math stack exchange sites. math.stackexchange.com/questions/tagged/probability looks quite active (and in fact there's the OP's question asked and answered) \$\endgroup\$ – Neil Slater Mar 31 '14 at 17:11

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