Eeny, meeny, miny, moe

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 \(a+b?\)

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