15. $A B C D$ is a rectangle with $A B=2$ and $B C=1$. A point $P$ is randomly selected on $C D$. Find the probability that $\angle A P B$ is the largest among the three interior angles of $\triangle P A B$.
$A B C D$ 是長方形, 其中 $A B=2, B C=1$ 。現於 $C D$ 上隨意選一點 $P$ 。求 $\angle A P B$ 是 $\triangle P A B$ 三隻內角中最大的一隻的概率。
15. $\sqrt{3}-1$
15. Note that the requirement of the question is satisfied if and only if both $A P$ and $B P$ are smaller than 2 , or equivalently, both $D P$ and $C P$ are smaller than $\sqrt{3}$. Hence if $D C$ is part of the number line with $D$ representing 0 and $C$ representing 2, then $P$ must be between $2-\sqrt{3}$ and $\sqrt{3}$ in order to satisfy the condition, and the probability for this to happen is
$$
\frac{\sqrt{3}-(2-\sqrt{3})}{2}=\sqrt{3}-1 \text {. }
$$