Two uniforms in the same dyadic band

Let $X$ and $Y$ be independent and uniform on $(0,1)$. What is the probability that $\lfloor\log_2 X\rfloor$ and $\lfloor\log_2 Y\rfloor$ are equal?

$1/3$. The floor of $\log_2$ equals $-n$ exactly on the band $[2^{-n},2^{-n+1})$, which has length $2^{-n}$. Both variables share band $n$ with probability $4^{-n}$, and $\sum_{n\ge1}4^{-n}=\tfrac13$.