Sure, let me provide a corrected proof.

Let $x$ and $y$ be two random integers. The product $xy$ is even if and only if either $x$ is even or $y$ is even.

The probability that $x$ is even is $\frac{1}{2}$ and the probability that $x$ is odd is also $\frac{1}{2}$. By the same reasoning, the probability that $y$ is even is $\frac{1}{2}$ and the probability that $y$ is odd is $\frac{1}{2}$.

Therefore, the probability that either $x$ or $y$ is even is given by:

$\frac{1}{2} + \frac{1}{2} = 1$

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

I hope this correct proof satisfies your need. Let me know if you need anything else!
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