For $0\le x<1$, 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 $f(x)={\displaystyle \sum _{i=1}^{\mathrm{\infty }}\frac{d_i(x)}{i^2}}$.

Let $p(a)$ be probability that $f(x)>a$, 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.