<|im_start|>system
You are a helpful AI assistant.<|im_end|>
<|im_start|>user
Write a proof, using LaTeX, showing that choosing two random integers and multiplying them together results in an even number with probability $\frac{3}{4}$.<|im_end|>
<|im_start|>assistant
Here is a LaTeX source file containing the proof:
```latex
\begin{document}

\textbf{Statement:}
Choosing two random integers and multiplying them together results in an even number with probability $\frac{3}{4}$.\\

\textbf{Proof:}
Let's call the two randomly chosen integers $x$ and $y$. 
The probability that each is odd is $\frac{1}{2}$. 
However, we know that in order to get an odd number from multiplication, 
both multiplicands must be odd.
As such, the probability that the product is odd is $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$,
since the two random integers are independently chosen.
Due to the fact that all natural numbers are even or odd,
and the probability that the product is odd is $\frac{1}{4}$, 
we can infer that the probability that the product is even is $1 - \frac{1}{4}$, or $\frac{3}{4}$.
This completes the proof.

\end{document}
```<|im_end|>
<|im_start|>user
That's not correct, and your reasoning is entirely flawed. Your premise that a product is even if and only if both $x$ and $y$ are both even or both odd is wrong. Two odd numbers, when multiplied together, always result in an odd number. A product $xy$ is even if either $x$ or $y$ is even, and that's how you can arrive at the $\frac{3}{4}$ probability. But I need a proof for this, so please formalize this fact in a proof.<|im_end|>
<|im_start|>assistant
Thank you for the correction, here is the proof using LaTeX:

Let $x$ and $y$ be two random integers. The product $xy$ is odd if and only if both $x$ and $y$ are odd.

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 $\frac{1}{2}$.

The probability that the product of $xy$ is odd is:

$\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$

The probability that the product of $xy$ is even is :

$1 - \frac{1}{4} = \frac{3}{4}$

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