A television network earns an average of $1.6 million each season from a hit pro

Question

A television network earns an average of $1.6 million each season from a hit program and loses an average of $400,000 each season on a program that turns out to be a flop, and of all programs picked up by this network in recent years, 25% turn out to be hits and 75% turn out to be flops.

Suppose that an actual (not perfectly reliable) market research report has the following characteristics based on historical data: if the program is actually going to be a hit, there is a 90% chance that the market researchers will predict the program to be a hit, and if the program is actually going to be a flop, there is a 20% chance that the market researchers will predict the program to be a hit.

Given this information, what are the posterior probabilities that a show will be a hit or a flop, given the market research report?

Please show calculations.

Explanation / Answer

Let A be the event that program is a hit

P(A) = 0.25

Let B be the event that report says program is a hit

P(B) = 0.25x0.90 + 0.75x0.20 = 0.375

P(A|B) = P(A and B)/P(B)

Here, P(A and B) = 0.25x0.9 = 0.225

P(A|B) = 0.225/0.375 = 0.6

Given the market research report, the posterior probability of the show being a hit is 0.6

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