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 $2\mathrm{\%}$ chance to score a total of exactly $20$ points.

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