9. For a real number $\alpha$, let $[\alpha]$ denote the greatest integer not exceeding $\alpha$. For example, $[3]=3,[\sqrt{2}]=1,[-\pi]=-4$. Let $x$ be a positive real number. If $\left[\log _{2} x\right]$ is even, then $x$ is called a lucky number. If a number is randomly selected from the interval $(0,1)$, the probability that it is a lucky number is $\qquad$.
Answer $\frac{1}{3}$.
Analysis Note that when $x \in(0,1)$, $\log _{2} x<0$; therefore $\left[\log _{2} x\right]$ is even if and only if
$$
\log _{2} x \in[-2,-1) \cup[-4,-3) \cup[-6-5) \cup \cdots
$$

which means $x \in\left[2^{-2}, 2^{-1}\right) \cup\left[2^{-4}, 2^{-3}\right) \cup\left[2^{-6}, 2^{-5}\right) \cup \cdots$, the sum of the lengths of these intervals is
$$
\frac{1}{2^{2}}+\frac{1}{2^{4}}+\frac{1}{2^{6}}+\cdots=\frac{1}{4} \cdot \frac{1}{1-\frac{1}{4}}=\frac{1}{3}
$$

Therefore, the probability that $x$ is a lucky number is $\frac{1}{3}$.