Eeny, meeny, miny, moe

Probability Level 4

An integer nn is chosen uniformly at random from the set {512,513,,1023}.\{512,513, \ldots, 1023\}. We take N,N, the binary expansion of nn and choose 2 of the digits at random. We change each of these digits, either from 00 to 11 or from 11 to 0,0, to create a new binary expansion M,M, which is the binary expansion for a number m.m. The probability that m>nm > n can be expressed as ab,\frac{a}{b}, where aa and bb are coprime positive integers. What is the value of a+b?a+b?

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