Suppose you are deciding whether or not to bring an umbrella to school and the weathercan be either sunny or rainy. Initially, you think the probability that it will be sunny isP (S) = 0.6, and the probability that it will be rainy is P (R) = 0.4. If you bring an umbrellaand it rains, your payoff is 6. If you bring an umbrella and it is sunny, your payoff is 9. Ifyou don’t bring an umbrella and it rains, your payoff is 0. If you don’t bring an umbrellaand 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 correct70% of the time. That is, the probability that the website says it will be sunny given that itactually will be sunny is 0.7, and the probability that the website says it will rain given thatit actually will rain is 0.7. Suppose that you suffer from confirmation bias. You initially thinkit 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 ownconfirmation bias. If your interpretation is that you have seen two sun forecasts, whatis your updated belief about the probability of sun? Will you bring your umbrella?
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. 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?
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