[math-fun] "Punctured" uniform discrete distributions

The discussion about g^X mod p made me start thinking about a uniform discrete distribution that had a small number of "holes" -- elements where the probability was zero. E.g., prob(y = g^x mod p, x in Z) = if y=0 then 0 else if 0<y<p then 1/(p-1) If one is sampling from a distribution X' with exactly one hole, where |X'|=n, how many samples would be required to start to distinguish between the uniform distribution X (|X|=n) with zero holes and the uniform distribution X' (|X'|=n) with one hole? By "start to distinguish", I mean "become suspicious" or "statistically significant".