Eeny, meeny, miny, moe

Probability Level 4

An integer $n$ is chosen uniformly at random from the set $\{512,513, \ldots, 1023\}.$ We take $N,$ the binary expansion of $n$ and choose 2 of the digits at random. We change each of these digits, either from $0$ to $1$ or from $1$ to $0,$ to create a new binary expansion $M,$ which is the binary expansion for a number $m.$ The probability that $m > n$ can be expressed as $\frac{a}{b},$ where $a$ and $b$ are coprime positive integers. What is the value of $a+b?$