I have the number 25. I am going to roll a D100 (1-100 inclusive) die. If the result of the roll is equal to or below that 25, I will add 1 to it, creating a 26 in its place. I will so this a total of 20 times using whatever the current number is; there is the potential for that number to remain at 25 or reach 45 after all 20 rolls, however unlikely.

What I'm after is a way to calculate the exact probability that, after all 20 rolls, the number will be 25, 26, 27, etc. to 45. I want the probability of each number in the range.

Just in case it's confusing, here's how it would look in practice:

*25, passed the roll; it's now a 26. It failed the second roll and remains a 26; it passed the next 3 rolls and became 29, but failed every single roll after, making the final result a 29. What was the probability of ending with this final result?*Anyone here know the solution? I don't expect you to do all of the calculations; I'm perfectly happy with a formula if you can help me get one.

I will also modify how many times we roll and the starting number. If possible, please identify those variables so I can play with them later to calibrate the system! Thanks for taking the time to consider this; I'm dying to hear what you think!