Proportion expected in a simulation

Question

X and Y are independent random variables uniform taking values ​​in the sets {1, 2, 3} and {1, 2}, respectively.

As can know by simulating the expected ratio for the pair $(X, Y)$ such that $X > Y$ is generate, for example, 2000 values.

Answer 1

The event $X\gt Y$ can happen in two disjoint ways: (i) $X=2$ and $Y=1$ or (ii) $X=3$ (and $Y$ is anything).

(i) The probability that $X=2$ and $Y=1$ is $\frac{1}{3}\cdot \frac{1}{2}$. This is $\frac{1}{6}$.

(ii) The probability that $X=3$ is $\frac{1}{3}$.

Add. The probability that $X\gt Y$ is $\frac{1}{6}+\frac{1}{3}$.