Suppose Dagwood (Blondie's husband) wants to eat a sandwich but needs to go on a diet. So, Dagwood decides to let the flip of a coin determine whether he eats. Using an unbiased coin, Dagwood will postpone the diet (and go directly to the refrigerator) if either (a) he flips heads on his first flip or (b) he flips tails on the first flip but then proceeds to get two heads out of the next three flips. Note that the first flip is not counted in the attempt to win two of three and that Dagwood never performs any unnecessary flips. Let Hi be the event that Dagwood flips heads on try i. Let Ti be the event that tails occurs on flip i.
(a) Sketch the tree for this experiment. Label the probabilities of all outcomes carefully.
(b) What are P[H3] and P[T3]?
(c) Let D be the event that Dagwood must diet. What is P[D]? What is P[H1|D]?
(d) Are H3 and H2 independent events?