Problem 286

Scoring probabilities

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.

得分概率

芭芭拉是一名数学家，也是一名篮球运动员。她发现，在距离x处投篮得分的概率恰好是(1 - ^x/_q)，其中q是一个大于50的常数。

在每次练习时，她都分别从距离为x = 1、x = 2、……、x = 50的地方投篮，根据她的记录，她恰好得分20次的精确概率为2%。

求出q，并将你的答案四舍五入到10位小数。