BAYES' THEOREM GOOD QUESTIONS

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Feb 20, 2024

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BAYES ’ T HEOREM GOOD QUESTIONS 1. Monty Hall Problem : Suppose that a new car is hidden behind one door on a game show, while goats are hidden behind two other doors. A contestant picks a door, and then (to heighten the suspense!) the host reveals what is behind a different door that he knows to have a goat. Then the host asks whether the contestant prefers to stay with the original door or switch to the remaining door. The question for students is: Does it matter whether the contestant stays or switches? If so, which strategy is better, and why? These questions will be explored below, but you can use this space to make a guess. Most people believe that staying or switching does not matter. An applet that allows students to play simulated games appears here . The following graph shows the results of 1000 simulated games with each strategy: It appears that switching wins more often than staying! We can determine the theoretical probabilities of winning with each strategy by using Bayes’ Theorem. More to the point, we can use our strategy of constructing a table of hypothetical counts. Suppose that the contestant initially selects door #1, so the host will show a goat behind door #2 or door #3. Let us use 300 for the number of games in our table, just so we’ll have a number that’s divisible by 3. Here is the outline of the table: How do we fill in this table? Let’s proceed as follows (I will start you on this): 1. Row totals: If the car is equally likely to be placed behind any of the three doors, then the car should be behind each door for 100 of the 300 games. 2. Bottom (not total) row: Remember that the contestant selected door #1, so when the car is actually behind door #3, the host has no choice but to reveal door #2 all 100 times. Finish filling in the table (rows 1 and 2) above and use it to explain how switching leads to a 2/3 probability of winning.

2. Imagine you are a member of a jury judging a hit-and-run driving case. A taxi hit a pedestrian one night and fled the scene. The entire case against the taxi company rests on the evidence of one witness, an elderly man who saw the accident from his window some distance away. He says that he saw the pedestrian struck by a blue taxi. In trying to establish her case, the lawyer for the injured pedestrian establishes the following facts: • There are only two taxi companies in town, “Blue Cabs” and “Green Cabs.” On the night in question, 85 percent of all taxis on the road were green and 15 percent were blue. • The witness has undergone an extensive vision test under conditions like those on the night in question and has demonstrated that he can successfully distinguish a blue taxi from a green taxi 80 percent of the time. Most people presented with this scenario placed a high probability on the witness being correct that the guilty taxi was blue. But a probability assessment reveals otherwise. In a quick intuitive assessment of the situation, respondents are likely to confuse the conditional probability that the witness said the taxi was blue given that it was blue (stated as 0.8) with the conditional probability that the taxi was blue given that the witness said it was blue ( which must be calculated ). Use a tree diagram AND a table to find the correct conditional probability: P(Taxi was blue | Witness said blue)

3. Super cool extension to the Monty Hall Problem (only for the brave - mental pygmies need not try this): Now suppose that the game show producers place the car behind door #1 50% of the time, door #2 40% of the time, and door #3 10% of the time. What strategy should you use? In other words, which door should you pick to begin, and then should you stay or switch? What is your probability of winning the car with the optimal strategy in this case? Explain. Hint remember the bottom line from above: By switching, you only lose if you were right to begin with. So, the optimal strategy here is to select door ____, and then ________________________________________________. Using this strategy you only lose if _____________________________________________. This optimal strategy gives you a _____chance of winning the car. One final thought: Why is Bayes’ Theorem so cool? It provides the mechanism for updating uncertainty in light of partial information, which enables us to answer important questions, such as the reliability of medical diagnostic tests (class example), being a more enlightened juror (#2 above) and also fun recreational ones, such as the Monty Hall Problem.

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