Problem 1: Amy is a private coach in a gym. In order to encourage her clients to come to her training sessions regularly and on time, she decides to establish a fund of $1,000 from which she will pay an amount, q, at the end of the year to each of her 20 clients who achieve perfect attendance over that year.
Suppose each client has a 1% chance of achieving perfect attendance during any given year, independent of any other client. Determine the maximum value of q such that there is a less than 1% chance that the $1,000 fund will be inadequate to cover all payments for perfect attendance.