A ratio between one and two

Idea. The pair $(q,p)$ is a uniform point in the unit square, so the probability is the area of the region $q<p<2q$.

Step 1: the region. It lies between the lines $p=q$ and $p=2q$ inside the square. The line $p=2q$ leaves the square at $(\frac{1}{2},1)$, so the region is the triangle with corners $(0,0)$, $(\frac{1}{2},1)$ and $(1,1)$.

Step 2: its area. The triangle has a horizontal side from $(\frac{1}{2},1)$ to $(1,1)$ of length $\frac{1}{2}$, and height 1 from that side down to the origin: $$\frac{1}{2}\cdot \frac{1}{2}\cdot 1=\frac{1}{4}.$$

So the probability is $\frac{1}{4}$.

By symmetry. The ratio is below 1 with probability $\frac{1}{2}$, and above 2 when $q<\frac{p}{2}$, a triangle of area $\frac{1}{4}$. What remains for the ratio between 1 and 2 is $1-\frac{1}{2}-\frac{1}{4}=\frac{1}{4}$.