Tom has a food truck, which he takes to events. He typically makes $20 in profit
ID: 3045823 • Letter: T
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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