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Scoring probabilities

Problem 286

Published on Saturday, 3rd April 2010, 05:00 am; Solved by 1180

Barbara is a mathematician and a basketball player. She has found that the probability of scoring a point when shooting from a distance x is exactly (1 - x/q), where q is a real constant greater than 50.

During each practice run, she takes shots from distances x = 1, x = 2, ..., x = 50 and, according to her records, she has precisely a 2 % chance to score a total of exactly 20 points.

Find q and give your answer rounded to 10 decimal places.