Conditional probability multiple conditions

Question

I have a question about conditional probability. Is it true that $P(A|B|C)$ equal to $P((A|B)|C)$ equal to $P(A|(B|C))$?

Answer 1

Yes, they are all identical.

If even one of A, B, or C is true, then all of the statements should evaluate to true. You can test that out for yourself. One of the OR's will become true, which makes the whole expression true.