Tom has a food truck, which he takes to events. He typically makes $20 in profit

Question

Tom has a food truck, which he takes to events. He typically makes $20 in profit for every 100 attendees at an event.

Tom 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, this event 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. Tom 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 a probability of .9. If the weather is not sunny, the report will predict sunny weather with a probability of .2.

What is the probability that the weather report predicts sunny weather?

Explanation / Answer

This solution is based on the total probability theorem.

P(A) = P(A B) + P(A B')

A = Report predicts sunny weather

B = Weather is actually sunny

B' = Weather is not actually sunny

P(A) = P(A | B) * P(B) + P(A | B')P(B') [By the Bayes' Theorem]

P(A | B) = P(Given that the weather is sunny, prob. of report predicting sunny weather) = 0.9

P(B) = P(Weather being sunny) = 60% = 0.6

P(A | B') = P(Given that the weather is not sunny, prob. of report predicting sunny weather) = 0.2

P(B') = P(Weather not being sunny) = 100% - 60% = 40% = 0.4

P(A) = 0.9(0.6) + 0.2(0.4) = 0.54 + 0.08 = 0.62

Answer: The probability that the weather report predicts sunny weather is 0.62

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