A large company may satisfy its fuel$-$oil

requirements either by one annual contract or a series of

separate monthly contracts throughout the winter. The

cost for the annual contract is $$3\mathrm{.}20$ per gallon. If the year

is $$normal$,$ the cost for the monthly contracts will

average $$3\mathrm{.}00$ per gallon, while if the year is a $$scarce$$

year with respect to fuel oil, the average cost of the

monthly contracts will be $$3\mathrm{.}80$ per gallon. The company

will use $100,000$ gallons during the year, and the manager

estimates a $1/10$ chance of a scarce year.

The manager may spend $$500$ to obtain a

professional economic forecast of whether the year will be

a normal or scarce one. Data on previous forecasts and

actual occurrences for the past $20$ years is shown: