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Martin's Service Station

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Martin's Service Station Martin is thinking about offering driveway-plowing services this winter. To do so, he'd need either to purchase a blade which could be mounted onto one of his tow-trucks (which would let him plow when the tow-truck wasn't needed for towing), or to buy a new snowplow vehicle dedicated purely to snow removal. How well his decision works out will depend, of course, on how heavy the snowfall is over the winter season. The payoff table ($) for Martin's Service Station follows: Snowfall Heavy Moderate Light s1 s2 s3 Decision Alternatives Purchase snowplow d1 7000 2000 -9000 Do not invest d2 0 0 0 Purchase blade d3 3500 1000 -1500 Probability of States of Nature 0.4 0.3 0.3 Suppose Martin decides to wait to check the September temperature pattern before making a final decision.
Estimates of the probabilities associated with an unseasonably cold September (I1) are P(I1|s1) = .30,
P(I1|s2)= .20, P(I1|s3) = .05. If Martin observes an unseasonably cold September, what is the recommended decision? If Martin does not observe an unseasonably cold September (I2), what is the recommended decision? snowfall heavy moderate light snowplow $7,000 $2,000 -$9,000 decision nothing $0 $0 $0 blade $3,500 $1,000 -$1,500 likelihood 40.00% 30.00% 30.00% Pr( September | snowfall ) heavy moderate light likelihood Sept. cold 30.00% 20.00% 5.00% 19.50% Sept. mild 70.00% 80.00% 95.00% 80.50% Pr( snowfall | September ) heavy moderate light Sept. cold 61.54% 30.77% 7.69% Sept. mild 34.78% 29.81% 35.40% E[ profit ] Sept. cold Sept. mild snowplow $4,230.77 -$155.28 nothing $0.00 $0.00 blade $2,346.15 $984.47 maximum $4,230.77 $984.47 $1,617.50

Martin should buy the snowplow if September is cold, and only the blade if September is mild. This yields him an expected profit of $1,617.50.

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