Confusion About Min-Cut Probabilities
I don't understand what you mean by "$(A_i,B_i) = ((A_1,B_1),\ldots,(A_t,B_t))$", an obviously false statement. Perhaps you meant $(A_i,B_i) \in \{(A_1,B_1),\ldots,(A_t,B_t)\}$?

I don't quite understand your interpretation of the statement $P((A_i,B_i)) \geq p := 1/\binom{n}{2}$. Here is the correct interpretation:


  For any minimum cut $C$, the probability that Karger's algorithm outputs $C$ is at least $p := 1/\binom{n}{2}$.


This is exactly what A states.

For B, you need to give an example of a graph which satisfies $P(C) \leq 1/\binom{n}{2}$ for all cuts $C$. One such example is a cycle, an example you were probably shown in class.