7. On a line segment $O A$ of length 1, two points $B$ and $C$ are chosen randomly. Let the length of segment $O B$ be $b$, and the length of segment $O C$ be $c$. Then the probability that $[\ln b]=[\ln c]$ is $\qquad$
7. $\frac{\mathrm{e}-1}{\mathrm{e}+1}$.

Notice that, the point $(b, c)$ satisfies the condition if and only if $\ln b, \ln c \in [k, k+1)$, i.e., $b, c \in \left[\mathrm{e}^{k}, \mathrm{e}^{k+1}\right)$.

Thus, these points must all lie within the square region $\left[\mathrm{e}^{k}, \mathrm{e}^{k+1}\right) \times \left[\mathrm{e}^{k}, \mathrm{e}^{k+1}\right)$.
Therefore, the area of all points that satisfy the condition is
$$
\begin{array}{l}
\sum_{k=1}^{+\infty}\left(\mathrm{e}^{-k+1}-\mathrm{e}^{-k}\right)^{2}=(\mathrm{e}-1)^{2} \sum_{k=1}^{+\infty} \mathrm{e}^{-2 k} \\
=(\mathrm{e}-1)^{2}\left(\frac{1}{1-\mathrm{e}^{-2}}-1\right)=\frac{\mathrm{e}-1}{\mathrm{e}+1} .
\end{array}
$$