Question: Jenny has a food truck, which she takes to events. She typically makes

Question

Question:

Jenny has a food truck, which she takes to events. She typically makes $20 in profit for every 100 attendees at an event. Jenny is interested in reserving a space at a big outdoor event. If the weather is good, this event attracts 100,000 people. If the weather is poor, it only attracts 20,000 people. Reserving a space costs $6,000. Based on historic weather reports, there is a 60% chance the weather will be sunny. Jenny has the option of paying $500 to get a more accurate weather report: If the weather will be sunny, the report will predict sunny weather with probability 0.9. If the weather is not sunny, the report will predict sunny weather with probability 0.2.

Find:

How much would perfect information (a psychic's foretelling of the future) about the weather be worth to Tom? Answer in units of dollars.

Explanation / Answer

For good weather, the profit is $20 * (100,000 / 100) = $20,000

For bad weather, the profit is $20 * (20,000 / 100) = $4,000

Payoff for good wether = $20,000 -  $6,000 = $14,000

Payoff for bad wether = $4,000 -  $6,000 = - $2,000

The Payoff table is,

Expected monetary value (EMV) for Reserve Space = 14,000 * 0.6 - 2,000 * 0.4 = 7,600

Let S be the event that weather is Sunny. Let R be the event that report predicts weather be Sunny, Then,

P(R | S) = 0.9

P(R | ~S) = 0.2

By law of total probability,

P(R) = P(R | S) P(S) +  P(R | ~S) P(~S)

= 0.9 * 0.6 + 0.2 * 0.4 = 0.62

By Bayes theorem,

P(S | R) = P(R | S) * P(S) / P(R)

= 0.9 * 0.6 / 0.62 = 0.871

So, given the report predicts Sunny weather, the weather actually is sunny is 0.871

Expected value with prefect information = 0.871 * 14,000 - (1 - 0.871) * 2,000 = 11,936

Expected value without prefect information = EMV = 7,600

Expected value of Perfect information = Expected value with prefect information - Expected value without prefect information

= 11,936 - 7,600 = 4,336

So, $4,336 of perfect information about the weather be worth to Tom.

Alternatives Weather Good(Sunny) Poor Reserve Space 14,000 -2,000 Do not reserve Space 0 0

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