The Monty Hall Problem is named for the long-time host of the game show Let's Make...
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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. 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.
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