I read the other answers in the thread, and I definitely would have gotten this wrong without that. This answer stands a chance at least
Independence
A and C are independent, A and B are not. Since you have a chance to win on turn 4 that you do not have on turn 1, P(A|B) = 1/2 + 1/16 + 1/64… = 7/12 != P(A). Conversely, P(A|C) = 1/4 + 1/16 + 1/64… = 1/3 = P(A)
response
Thanks for the catch! Once I noticed that the probability of winning on the initial flip was the difference between the two, I stopped thinking about the other terms…