1. Suppose that A and B are independent events. Show that Ac and Bc are independent events.
2. There are three cards. The ?rst is green on both sides, the second is red on both sides and the third is green on one side and red on the other. We choose a card at random and we see one side (also chosen at random). If the side we see is green, what is the probability that the other side is also green? Many people intuitively answer 1/2. Show that the correct answer is 2/3.
3. Suppose that a fair coin is tossed repeatedly until both a head and tail have appeared at least once.
(a) Describe the sample space ?.
(b) What is the probability that three tosses will be required?