## Scoring probabilities

### Problem 286

Published on Saturday, 3rd April 2010, 05:00 am; Solved by 1372; Difficulty rating: 45%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.