Wayne County Math Teachers’ Circle One­day August Immersion, Friday, August 12, 2016 Monty Hall Problem 1. Suppose you flip a coin, and we denote getting heads as H and getting tails as T. a) What is the probability that you get a tail if you flip a coin once? b) List all outcomes when a coin is flipped twice. (For example, HT stands for head, tail). c) If you flip a coin twice, what is the probability of getting T on the second flip? d) List all outcomes when a coin is flipped three times. e) If you flip a coin three times, what is the probability of getting TTH? HTH? HHH? what is the probability of getting two heads? what is the probability that your second flip is H? f) Without listing all the outcomes, what is the probability of getting HTTHT after 5 flips? What is the probability that the first two of the five flips are heads? 2. The M
onty Hall Problem gets its name from a TV game show, Let's Make A Deal, that was hosted by Monty Hall. The set up is as follows: there are three closed doors in front of you, and behind one door is a car while behind the other two are goats. You are given the opportunity to choose one of the doors. Monty Hall knows where the car is, and, once you have made your choice of door, he will open one of the remaining doors to reveal a goat. You then have the choice to switch your choice of door, or to stay with your original choice. In order to maximize the chances of winning the prize, do you switch or not? Discuss and make a decision. Wayne County Math Teachers’ Circle One­day August Immersion, Friday, August 12, 2016 Monty Hall Problem 3. Before analyzing the Monty Hall problem, let’s do what mathematicians often do: analyze a seemingly more difficult problem. Instead of three doors, imagine that there are 100 doors. Behind one door is a car while behind the other 99 doors there are goats. You are given the opportunity to choose one of the doors. Monty Hall knows where the car is, and, once you have made your choice of door, he will open 98 of the remaining doors to reveal 98 goats. There are now 2 closed doors remaining, do you switch or not? Some questions to consider: a) What is the probability of the car being behind the door you chose first? b) What is the probability of the car being behind the other 99 doors? (Think of the 99 doors as a single group.) c) After elimination of 98 wrong doors, what is the probability of the door you originally chose being the door hiding the car? d) What is the probability of the prize being behind the other door? (Think of the 99 doors as a group.) e) Stay or switch? Which door has the bigger probability of winning? f) Do you switch or stay with the original pick when there are 50 doors instead? Why or why not? 4. Now analyze the original three door Monty Hall problem. Stay or switch? Which door has the bigger probability of winning? 5. When a coin is flipped three times, which is more likely: HHH or THH? A more subtle question: We repeatedly flip a coin until either the sequence HHH or THH occurs. Which is more likely to occur first? 6. An even more subtle question: We repeatedly flip a coin until either the sequence HHT or THH occurs. Which is more likely to occur first? 7. Show that HHT is more likely to occur before HTH and TTH is more likely to occur before THH. Conclude that no matter which length three sequence of Hs and Ts your partner chooses, you can find a sequence which is at least twice as likely to occur first.