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 $x = 1, x = 2, \dots, x = 50$ 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.
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