What are the Odds

Let $\psi$ be the obtuse angle, $\theta$ be the other two angles and let $s$ be the third side length. Fold the triangle in half along its axis of symmetry. The result is a right triangle with complementary angles $\frac{\psi}{2}$ and $\theta$ , and with base length $\frac{s}{2}$ and hypotenuse $32$ .

The angle $\psi$ is obtuse, so $\frac{\pi}{4} < \frac{\psi}{2} < \frac{\pi}{2}$ which means that $0 < \theta < \frac{\pi}{4}$ . Thus, $\frac{1}{\sqrt{2}} < \cos{\theta} < 1$ . We also know $32\cos{\theta} = \frac{s}{2}$ , or $\cos{\theta} = \frac{s}{64}$ . Therefore, $\frac{1}{\sqrt{2}} < \frac{s}{64} < 1$ , or $\frac{64}{\sqrt{2}} < s < 64$ . There are 18 integers strictly between $\frac{64}{\sqrt{2}}$ and $64$ , 4 of which are prime. So the probability that $s$ is prime is 2 in 9.