Problem - 1650
Integers $a, b, c,$ and $d$, not necessarily distinct, are chosen independently and at random from $0$ to $2019$, inclusive. What is the probability that $(ad-bc)$ is even?
The value of $(ad-bc)$ will be even if and only if $ad$ and $bc$ have the same parity. We note that there are same number of odd and even integers within the give range. As such, there are $\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$ chance that $ad$ is odd and $\frac{3}{4}$ chance that $ad$ is even. The same probability applies to $bc$. Therefore the result is $$\frac{1}{4}\times\frac{1}{4}+\frac{3}{4}\times\frac{3}{4}=\boxed{\frac{5}{8}}$$