jmc

Since $\angle APB=90^{\circ }$ if and only if $P$ lies on the semicircle with center $(2,1)$ and radius $\sqrt{5}$, the angle is obtuse if and only if the point $P$ lies inside this semicircle. The semicircle lies entirely inside the pentagon, since the distance, 3, from $(2,1)$ to $\overline{DE}$ is greater than the radius of the circle. Thus the probability that the angle is obtuse is the ratio of the area of the semicircle to the area of the pentagon.

Let $O=(0,0)$, $A=(0,2)$, $B=(4,0)$, $C=(2\pi +1,0)$, $D=(2\pi +1,4)$, and $E=(0,4)$. Then the area of the pentagon is $$[ABCDE]=[OCDE]-[OAB]=4\cdot (2\pi +1)-\frac{1}{2}(2\cdot 4)=8\pi ,$$and the area of the semicircle is $$\frac{1}{2}\pi (\sqrt{5}{)}^2=\frac{5}{2}\pi .$$The probability is $$\frac{\frac{5}{2}\pi }{8\pi }=\boxed{\frac{5}{16}}.$$