Problem 1.3 (Video 1.3, 1.4, Lecture Problem) Thor is trying to determine Loki's recent whereabouts. His approach is to go to a random planet and ask the local leader, \"Has Loki visited this planet recently?\" to which they will answer either \"Yes\" or \"No.\" There is one problem: some of the planets are secretly controlled by Loki and the leaders will lie to protect him. The probability that a planet is controlled by Loki is 1/3. The probability that Loki has visited a planet recently is 1/2 if he controls it and 1/4 if he does not control it. If the planet is controlled by Loki, the leader will lie with probability 3/4. If the planet is not controlled by Loki, a leader will lie with probability 1/5 (for their own reasons that have nothing to do with Loki). (a) Thor picks a random planet and asks its leader if Loki has visited recently. Draw and label a tree diagram of all the probabilities and conditional probabilities leading up to the leader's answer. Specifically, let C be the event that Loki controls the planet, V be the event that Loki has visited this planet recently, and Y be the event that the leader answers \"Yes.\" (You can think of the answer \"No\" as Y.) Draw the tree with the sample space {2 at the root, followed by C, then V, and then Y. (b) Determine the probability that Loki does not control the planet and Loki has visited recently and the leader answers \"Yes,\" P[CNV NY]. (c) Determine the probability that the leader answers \"Yes\" given that Loki has visited recently, P[YV]. (d) Determine the probability that the leader answers \"Yes,\" P[Y]. (e) Determine the probability that Loki has visited recently given that the leader answers \"Yes,\" PV|Y]. (f) After some careful observation, Thor manages to identify a planet that Loki does not control. Given that the leader answers \"Yes\" on this planet, what is the probability that Loki has visited recently, P[V|Y N C*]? - Cavny 2 C Govnye yo Covent . Gavenye WIN 4 Co . cenvnye s/w conve oconveny . Genvenue