Matching floors of a logarithm

Let $X, Y \overset{\text{iid}}{\sim} \text{Uniform}(0,1)$ . Compute $\mathbb{P}\!\left[\lfloor \log_2 X \rfloor = \lfloor \log_2 Y \rfloor\right]$ .

Show hints (2)+

$\lfloor\log_2 X\rfloor=-k$ means $X\in[2^{-k},2^{-k+1})$ , an interval of length $2^{-k}$ .
Both variables share band $k$ with probability $(2^{-k})^2$ . Sum the geometric series over $k\ge1$ .

Answer

Reveal answer →

0.3333 (± 0.001)

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