Probability that 3 randomly selected elements of a set are equal check I have the following question:
Let A = {1, 2, 3, . . . , 100}. Let x, y, and z be elements in A that are chosen independently
and uniformly at random. What is the probability that x = y = z?
Because if x is chosen first, only y and z have to be equal to it, would this mean that the answer is $\frac{1}{100 \cdot 100}$? 

That depends on whether $x, y, z$ are necessarily distinct (chosen without replacement).  If they are necessarily distinct, then the answer is (EDIT) $0$, obviously.
If they are not necessarily distinct (chosen with replacement), then your answer of $1/100^2 = 0.0001$ is correct.
