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: http://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.)