3. Let $[x]$ denote the greatest integer not exceeding the real number $x$. 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. Then the probability that a number randomly selected from the interval $(0,1)$ is a lucky number is
3. $\frac{1}{3}$.

Notice that when $x \in (0,1)$, $\log _{2} x < 0$. Therefore, $\left[\log _{2} x\right]$ is even if and only if
$$
\begin{array}{l}
\log _{2} x \in \bigcup_{k=1}^{\infty}[-2 k, 1-2 k) \\
\Leftrightarrow x \in \bigcup_{k=1}^{\infty}\left[2^{-2 k}, 2^{1-2 k}\right) .
\end{array}
$$

The sum of the lengths of these intervals is
$$
\sum_{k=1}^{\infty} \frac{1}{2^{2 k}}=\frac{1}{4} \times \frac{1}{1-\frac{1}{4}}=\frac{1}{3} \text {. }
$$

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