Regarding these questions based on the pay-off table, I would like to know how to draw out posterior possibility like the solution mentioned above.

27. Fenton and Farrah Friendly, husband-and-wife car dealers, are soon going to open a new dealership. They have three offers: from a foreign compact car company, from a U.S. producer of full-sized cars, and from a truck company. The success of each type of dealership will depend on how much gasoline is going to be available during the next few years. The profit from each type of dealership, given the availability of gas, is shown in the following payoff table: Gasoline Availability Shortage .6 Surplus .4 Dealership $ 300,000 $150,000 Compact cars Full-sized cars Trucks - 100,000 120,000 600.000 170,000 50. The Friendlys in Problem 27 are considering hiring a petroleum analyst to determine the future availability of gasoline. The analyst will report that either a shortage or a surplus will occur. The probability that the analyst will indicate a shortage, given that a shortage actually occurs is .90; the probability that the analyst will indicate a surplus, given that a surplus actually occurs is .70. a. Determine the decision strategy the Friendlys should follow, the expected value of this strategy, and the maximum amount the Friendlys should pay for the analyst's services. b. Compute the efficiency of the sample information for the Friendly car dealership. 50. a) Let s = shortage: st = surplus; P(s) = .6; P(s) = .4. Let S = report of shortage: st = report of surplus; P(STS) = 90; P(SIS) = 10; P(S*s*) = .70; P(S1st) = .30. P(S) = .66 = .818 P(s|) = 1 -.818 = .182 Pst st)P(s) Pst S)= P(sts)P(s)+P( Ss)P(s) (.70)(.40) (.70)(.40) +(.10).60) P(S) = .34 = .824 P(5 st) = 1-.824 = .176 P(S 5 )P(s) P(S|S )P(S) + P( SS)P(s) (90)(-60) (90)(.60)+(30).40) P( SS) = $272,700 Pls" IS) = .818 $300,000 Compacts $150,000 P(SIS) = .182 $27,400 PS IS ) = .818 $272,700 $100,000 5 Full-sized $600,000 P(s IS") = .182 P(S IS ) = .818 Trucks Shortage report P(S) - .66 $120,000 6 $170,000 $129,100 Pls IS) = .182 $342,094 $176,400 P(s"IS") = .176 $300,000 7 P(S) = .34 Surplus report Compacts $150,000 P(s+IS+) = .824 $476,800 P(S IS") = 176 -$100,000 3 8 Full-sized $600,000 $476,800 P(s'IS') = .824 Pls IS*) = .176 Trucks $120,000 9 $170,000 $161,200 P(s*IS') = .824