An easy way to do the odds analytically is to calculate the average expected number of trials needed for a success, and then note that the overall average success rate is the inverse of that.

E.g. in the author's example, where the rate starts at 5% and increases by 5% with each failure, the expected number of trials needed for a success would be:

    sum = 1 * 0.05 +                // hit the first trial
          2 * 0.95 * 0.10 +         // missed first, hit second
          3 * (0.95*0.9) * 0.15 +   // missed first two, hit third
          // ..and so on

Summing up that loop gives 5.29 for the expected number of trials, the inverse of which is 18.9%, matching the author's observed result.