You are correct.
The following is the updated proof using LaTeX:

Let $x$ and $y$ be two random integers. There are 4 different cases in which the multiplication between the two integers can be carried out:

$$
xy=
\begin{cases}
x_{odd} \cdot y_{odd} = xy_{odd}\\
x_{odd} \cdot y_{even} = xy_{even}\\
x_{even} \cdot y_{odd} = xy_{even}\\
x_{even} \cdot y_{even} = xy_{even}
\end{cases}
$$

Out of these 4 different cases, there are 3 different scenarios in which the product of $x$ and $y$ is an even number.

Thus, the probability that $xy$ is even is $\frac{3}{4}$
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