How much is the probability of $\cos x \cos(4-x)<0$ for $0<x<4$?

Question

I have this function $$f(x)=g(x)h(x)=\cos x \cos(4-x)$$ How much is the probability of $f(x)<0$ for $0<x<4$?

Answer 1

The rest of the answer assumes that $x$ is chosen uniformly over $[0,4]$.

$$\cos x\cos (4-x) > 0$$ for $$x\in \left(\frac{\pi}{2}, 4-\frac{\pi}{2}\right)$$ on $[0,4]$.

So the probability you require is $$1 - \frac{4-\pi}{4} = \frac{\pi}{4}\approx 0.785$$

For reference, here is the graph on $[0,4]$ -