How can I approach questions like this?
Problem:
Let $S$ be a finite set of random positive integers. What is the probability that the sum of $N$ randomly selected integers from this set $S$ is even?
Does the answer depend on $N$ (i.e., depending on whether $N$ is even or odd) or is it always equal to 1/2.
Best Answer
You can work $\pmod 2$ as you only care about even or odd. If you are asking about the probability for a randomly selected set $S$, then the probability is in fact $\frac 12$. In fact the set $S$ is a red herring-you can just think about flipping a coin, with heads giving an even number and tails giving an odd one. If $N=0$, the probability of an odd number of tails is $0$. But if $N$ is any greater, the chance of an odd number of tails is $\frac 12$, which you can prove by induction.