Question 3. (18 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 , and the probability that it will be rainy is . 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.

1.(9 Points) Suppose that you check two websites and see that both say it will be sunny. 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. If you are a Bayesian updater, what is your updated belief about the probability that it will be sunny? Will you bring your umbrella?

2.(9 Points) Now suppose that you suffer from confirmation bias. You are pretty sure that it is going 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 that it will be sunny? Will you bring your umbrella?