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ECON30019 Assignment 2
Due at 11:59 pm on Aug 27, 2024
Question 1. (25 points)
Charlie has made an investment in the stock market. Initially, he believes that the economy is “Good.” So he believes that each day the value of his investment goes up with probability 80% and goes down with probability 20%. Assume the probability of “up” or “down” on each day is independent.
1. (5 points) According to Charlie’s initial belief, what is the probability of having 5 straight days on which his investment goes up?
2. (20 points) After doing some research and reading some news, Charlie starts to won-der if maybe his initial belief about the economy was too optimistic. He still thinks the economy is “Good” with 70% confidence, but acknowledges a 20% chance that the economy is “OK” and a 10% chance that the economy is “Bad.” Charlie believes that in an “OK” economy, each day the value of his investment goes up with probability 60% or goes down with probability 40%. He believes that in a “Bad” economy, each day the value of his investment goes up with probability 10% or goes down with probability 90%.
Over the next 5 days, Charlie observes his investment goes up, down, up, down, down. After these observations, what does Charlie now believe is the probability that the economy is “Good?”
Question 2. (30 points)
In a base-rate neglect experiment, subjects are randomly divided into two groups, namely A and B.
• Subjects in Group A are told that Max has been randomly drawn from a population of 95% nurses and 5% doctors.
• Subjects in Group B are told that Max has been randomly drawn from a population of 5% nurses and 95% doctors.
• All subjects from both groups are provided with the following description about Max: Max lives in a beautiful house and drives an expensive car; he invests a lot of time in his career.
• All subjects from both groups are told that the probability that Max fits the description given that he is a doctor is 80%, and the probability that Max fits the description given that he is a nurse is 20%.
All subjects from both groups are then asked to estimate the probability that Max is a doctor given that he fits the description.
1. (12 points) Suppose subjects in Group A correctly use the probability theory. What should be their estimate?
2. (12 points) Suppose subjects in Group B correctly use the probability theory. What should be their estimate?
3. (6 points) Explain why this experiment can test base-rate neglect. What type of re-sponses given by the two groups may imply that they exhibit base-rate neglect?
Question 3. (45 points)
Suppose you are deciding whether or not to bring an umbrella to school and the weather can be either sunny or rainy. Initially, you think the probability that it will be sunny is P(S) = 0.6, and the probability that it will be rainy is P(R) = 0.4. If you bring an umbrella and it rains, your payoff is 6. If you bring an umbrella and it is sunny, your payoff is 9. If you don’t bring an umbrella and it rains, your payoff is 0. If you don’t bring an umbrella and it is sunny, your payoff is 10. Suppose that you check two websites for weather forecasts. You know that the two forecasts are independent of each other, and each forecast is correct 70% of the time. That is, the probability that the website says it will be sunny given that it actually will be sunny is 0.7, and the probability that the website says it will rain given that it actually will rain is 0.7.
1. (17 points) Suppose that you think you have seen two sun forecasts. You believe that you always correctly interpret all forecasts. If you are a Bayesian updater, what is your updated belief about the probability that it will be sunny? Do you bring your umbrella?
2. (23 points) Now suppose that you suffer from confirmation bias. You initially think it is more likely to be sunny, so you misinterpret rain forecasts as sun forecasts 20% of the time but always correctly interpret sun forecasts. You are aware of your own confirmation bias. If your interpretation is that you have seen two sun forecasts, what is your updated belief about the probability of sun? Will you bring your umbrella?
3. (5 points) Compare your answers in (3.1) and (3.2), what have you learned? Provide a brief discussion.