The "Music practice" problem
I have little knowledge of math in general, so I don't even know if this problem is meaningful or interesting, nor do I know how to search for similar problems.
Starting with 0, you either add or subtract 1 depending on a coin flip - let's say heads is +1, tails is -1. (You are not allowed to go into the negatives, you stick to zero in that case). The first coin flip has an 50-50% chance of being either heads or tails. If it is heads, you "weight" the coin in such a way that the chance of getting heads (+1) increases by 10%. If you get tails, you "weight" the coin in such a way that the chance of getting tails (-1) is increased by 5%. The "weighting" cannot be less than 5% or more than 95% (if that is the case, you just stick to the boundary you reached for the next flip. What is the probability of the total sum being at least 10 after 100 coin flips? (All numbers may be interchanged for symbols for generality).
(The original context, and rather redundant phrasing:
Anna is practicing a difficult part of a piece on her musical instrument by repeatedly playing through it. Her method consists of putting 10 matches on one side of the music stand, and moving them to the right one-by-one for each perfect playthrough of the music. However, if she makes a mistake, she moves one of the matches on the right (if there are any) back to the left. She considers the practice session a success if she manages to move all 10 matches to the right.
In the beginning, she has a 50 percent chance of the next playthrough being perfect. For each sucessful playthrough, the probability of the next playthrough being perfect increases by 10%, and for each failure, it decreases by 5%. The probability cannot go below 5% or above 95% (if it would, it just stays on the respective limit for the time).
What is the probability that she has moved all 10 matches by the 20th - 50th - 100th playthrough? What is the expected number of playthroughs by which she would succeed? What about a general case for m matches, p initial percent of success, and q and r percent changes to the probability?)