A ball chosen at random has only a 5% chance of coming from the lowest-numbered 5% of balls, and a 95% chance of coming from the other, higher-numbered balls. We are therefore 95% certain that our ball, number 7, has a higher number than the lowest 5% of balls. So we are 95% certain that the lowest-numbered 5% consists of fewer than 7 balls. If there are only 6 or fewer balls in a 5% sample, then the total number of balls must be (6*20)=120 or fewer. When we draw ball number 7, we are therefore immediately 95% confident that there are 120 or fewer balls in the urn.

Grant Hutchison