I.
You must take options A, B, or do nothing. If x happens, the payoff is 10 for A and 15 for B; if y happens, it is 30 for A and 40 for B; if z happens, it is -5 for A and -10 for B. Pr(x)=.6; Pr(y)=.3; Pr(z)=.1. If you do noting, the payoff is -10 if x happens, and 20 otherwise. What should you do?
II.
The National Forest Service is considering control-burning a tract of forest to eliminate brush and debris that could result in an uncontrolled fire with a damage of 100. The probability of such a fire is estimated to be 20%. No uncontrolled fire in such conditions has payoff 20. The control burning could be carried out with or without ditching (bulldozing a clear tract at the intended fire boundaries to reduce the chance of accidental propagation). Ditching costs 10. The payoff of a controlled fire is +80; that of an uncontrolled fire is -60 (it’s not -100 because of the presence of firefighters). The probability that the fire will uncontrollably spread without ditching is .5; that of an uncontrolled fire with ditching is.2. What should the National Forest Service do?
III.
You are the owner of a small oil company who has a lease on a parcel of land. You believe that the probability you strike oil is 30%. If oil is present you can sell the lease for 500 or drill it yourself for a cost of 20; if there is no oil, you will sell the lease for 5. If the market is good you can sell the oil you produced for +800; if it is bad, you can sell the oil for +500; if it is average, you can sell for 600. The probabilities of the market being good, average, and bad are .5, .3, and .2 respectively. You could pay 30 for a seismic survey. Given the presence of oil, the survey will report it 90% of the times; if no oil is present, the seismic survey will report its absence 70% of the times. What should you do?