You generate a random $N$-bit binary string, and compute $X = \Sigma_1^N x_i$, where the $x_i$ are the $0$ and $1$ entries of the string.
What is the probability that $X$ is odd, if $N$ is odd?
What is the probability that $X$ is odd, if $N$ is even?
I've been told that the first part is easy, but can't seem to figure out even that. I've tried figuring out the probability for $N-1$ to be odd as a starting point, but can't figure out even that.
Hint: Whatever the first $N-1$ bits are, the probability that $X$ is odd is $\frac{1}{2}$.