Binary Series

 Published on Sunday, 17th November 2019, 04:00 am; Solved by 202;
Difficulty rating: 60%

Problem 689

For $0 \le x \lt 1$, define $d_i(x)$ to be the $i$th digit after the binary point of the binary representation of $x$.
For example $d_2(0.25) = 1$, $d_i(0.25) = 0$ for $i \ne 2$.

Let $f(x) = \displaystyle{\sum_{i=1}^{\infty}\frac{d_i(x)}{i^2}}$.

Let $p(a)$ be probability that $f(x) \gt a$, given that $x$ is uniformly distributed between 0 and 1.

Find $p(0.5)$. Give your answer rounded to 8 digits after the decimal point.