This is a question from Bayesian Data Analysis by Andrew Gelman. 3rd Edition

Chapter 9 Exercise 2

Conditional probability and elementary decision theory: Oscar has lost his dog; there is a 70% probability it is in forest A and a 30% chance it is in the forst B. If the dog is in forest A and Oscar looks there for a day, he has a 50% chance of finding the dog. If the dog is in forest B and Oscar looks there for a day, he has an 80% change of finding the dog.

• If Oscar can search only one forest for a day, where should he look to maximize his probability of finding the dog? What is the probability that the dog is still lost after the search?
• Assume Oscar made the rational decision and the dog is still lost (and is still in the same forest as yesterday). Where should he search for the dog on the second day? What is the probability that the dog is still lost at the end of the second day?
• Again assume Oscar makes the rational decision on the second day and the dog is still lost (and is still in the same forest). Where should he search for the dog on the third day? What is the probability that the dog is still lost at the end of the third day?
• (Expected value of additional information.) You will now figure out the expected value of knowing, at the beginning, which forest the dog is in. Suppose Oscar will search for at most three days, with the following payoffs: -1 if the dog is found in one day, -2 if the dog is found on the second day, and -3 if the dog is found on the third day, and -10 otherwise.
• -what is Oscars expected payoff without the additional information?
• -what is Oscars expected payoff if he knows the dog is in forest A?
• -what is Oscars expected payoff if he knows the dog is in forest B?
• -Before the search begins, how much should Oscar be willing to pay to be told which forest his dog is in?