The Monty Hall Problem is named for the long-time host of the game show Let's Make a Deal. This was a simple-seeming problem that confounded many people in the 1970's - including some with PhD's! The simple version goes as follows: you are shown three doors; behind two of the doors are goats and behind the other is a car. If you correctly guess the door with a car, you win the car. You choose a door (say Door #1). Monty Hall then opens one of the two remaining doors say, Door #3 - revealing goat. You can either keep Door #1 or switch to Door #2 to try to get the car. (a) Use Bayes' Theorem to decide what to do: switch or stay? Hint: Let O; be the event "Monty opens Door i to reveal a goat." Let C; be the event "Car is behind Door i". Assume P(0) = 0, Monty only picked a door that has a goat behind it (from two choices of doors, #2 and #3), and P(C) = for all i (b) Describe what you did in part (a) using a tree diagram approach: First, construct a tree with two layers and explain why you can do so in this way. Then, fill in the probabilities at each branch and at the leaves.