7. If three points are randomly taken on a circle, the probability that the triangle formed by these three points is an acute triangle is $\qquad$

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7. $\frac{1}{4}$.

Assume $\triangle A B C$ is any inscribed triangle in a circle with radius 1, and the arc lengths opposite to $\angle A$ and $\angle B$ are $x$ and $y$, respectively. Then,
$$
\left\{\begin{array}{l}
0<x<2 \pi, \\
0<y<2 \pi, \\
0<x+y<2 \pi .
\end{array}\right.
$$

This system of inequalities represents the region $\triangle P O Q$ in Figure 3 (excluding the boundaries), with an area of $S_{1}=2 \pi^{2}$.
If $\triangle A B C$ is an acute triangle, then $x$ and $y$ satisfy
$$
\left\{\begin{array}{l}
0<x<\pi, \\
0<y<\pi, \\
\pi<x+y<2 \pi .
\end{array}\right.
$$

This system of inequalities represents the region $\triangle D E F$ in Figure 3 (excluding the boundaries), where $D$, $E$, and $F$ are the midpoints of $O P$, $O Q$, and $P Q$, respectively, with an area of $S_{2}=\frac{1}{2} \pi^{2}=\frac{1}{4} S_{1}$.
Therefore, the required probability is $p=\frac{S_{2}}{S_{1}}=\frac{1}{4}$.