A New Car! Precalculus B Name: Date: It's 1963. You are a contestant on the popular new game show Let's Make a Deal, hosted by Monty Hall. In front of you are three closed doors, marked 1, 2, and 3. Monty Hall says, "Behind two of these doors is a goat. But behind one door is a new car! Which door do you choose?" You take a deep breath and choose a door, door 2. Your heart is pounding as you wait for Monty to open door 2, but instead he says, "I'm going to open a different door for you first." And he opens door 1. Behind it, you see a goat. So the car is behind either the door you chose, door 2, or the third door, door 3. "Now," Monty says to you, "do you want to change your mind? Or would you like to make a deal and stick with door 2?" 1. Would you change your mind? Why or why not? 2. Let's look at this closely. When the game started, when all the doors were closed, what was the probability (as a fraction) that the car was behind your door, door 2? 3. What is the probability, before the doors are opened, that the car is behind either door 1 or door 3? Keep in mind that probability of mutually exclusive events should add up to 1. 4. Once Monty Hall opens door 1 to reveal a goat behind it, what is the probability that the car is behind door 2, the one you selected? Hint: It is not 1/2!
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It’s 1963. You are a contestant on the popular new game show Let’s Make a Deal, hosted by Monty Hall. In front of you are three closed doors, marked 1, 2, and 3. Monty Hall says, ‒Behind two of these doors is a goat. But behind one door is a new car! Which door do you choose?‒ You take a deep breath and choose a door, door 2. Your heart is pounding as you wait for Monty to open door 2, but instead he says, ‒I’m going to open a different door for you first.‒ And he opens door 1. Behind it, you see a goat. So the car is behind either the door you chose, door 2, or the third door, door 3. ‒Now,‒ Monty says to you, ‒do you want to change your mind? Or would you like to make a deal and stick with door 2?‒ 1. Would you change your mind? Why or why not?
Donna D.
Consider the Monty Hall problem, except that Monty enjoys opening Door 2 more than he enjoys opening Door 3, and if he has a choice between opening these two doors, he opens Door 2 with probability p, where 1/2 ≤ p ≤ 1. To recap: there are three doors, behind one of which there is a car (which you want), and behind the other two of which there are goats (which you don’t want). Initially, all possibilities are equally likely for where the car is. You choose a door, which for concreteness we assume is Door 1. Monty Hall then opens a door to reveal a goat, and offers you the option of switching. Assume that Monty Hall knows which door has the car, will always open a goat door and offer the option of switching, and as above assume that if Monty Hall has a choice between opening Door 2 and Door 3, he chooses Door 2 with probability p (with 1/2 ≤ p ≤ 1). (a) Find the unconditional probability that the strategy of always switching succeeds (unconditional in the sense that we do not condition on which of Doors 2,3 Monty opens). (b) Find the probability that the strategy of always switching succeeds, given that Monty opens Door 2. (c) Find the probability that the strategy of always switching succeeds, given that Monty opens Door 3.
Supreeta N.
Consider the following variation of the Monty Hall problem, where in some situations Monty may not open a door and give the contestant the choice of whether to switch doors. Specifically, there are 3 doors, with 2 containing goats and 1 containing a car. The car is equally likely to be anywhere, and Monty knows where the car is. Let 0 ≤ p ≤ 1. The contestant chooses a door. If this initial choice has the car, Monty will open another door, revealing a goat (choosing with equal probabilities among his two choices of door), and then offer the contestant the choice of whether to switch to the other unopened door. If the contestant's initial choice has a goat, then with probability p Monty will open another door, revealing a goat, and then offer the contestant the choice of whether to switch to the other unopened door; but with probability 1 − p, Monty will not open a door, and the contestant must stick with their initial choice. The contestant decides in advance to use the following strategy: initially choose door 1. Then, if Monty opens a door and offers the choice of whether to switch, do switch. (a) Find the unconditional probability that the contestant will get the car. Also, check what your answer reduces to in the extreme cases p = 0 and p = 1, and briefly explain why your answer makes sense in these two cases. (b) Monty now opens door 2, revealing a goat. So the contestant switches to door 3. Given this information, find the conditional probability that the contestant will get the car.
Sri K.
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