Scientific research on popular beverages consisted of 60 studies that were fully

Question

Scientific research on popular beverages consisted of 60 studies that were fully sponsored by the food industry, and 40 studies that were conducted with no corporate ties. Of those that were fully sponsored by the food industry, 13 of the participants found the products unfavorable, 25 were neutral, and 62 % found the products favorable. Of those that had no industry funding 40 % found the products unfavorable 13% were neutral, and 47% found the products favorable. What is the probability that a participant selected at random found the products favorable? If a randomly selected participant found the product favorable what is the probability that the study was sponsored by the food industry? If a randomly selected participant found the product unfavorable, what is the probability that the study had no industry funding?

Explanation / Answer

Let F shows the event that study sponsored by food industry and N shows the event that study had no industry fundings. So we have

P(F) = 0.60, P(N) = 0.40

Let A shows the event that participants found product favorable, B shows the event that participants found product unfavorable and C shows the event that particpant found product neutral.

Here we have

P(A|F) = 0.62, P(B|F) = 0.13, P(C|F) = 0.25

P(A|N) = 0.47, P(B|N) = 0.40, P(C|N) = 0.13

By the law of total probability, the probability that a participants found the product favorable is

P(A) = P(A|F)P(F) + P(A|N)P(N) = 0.62 * 0.60 + 0.47 * 0.40 = 0.56

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By the Baye's theorem, the probability that the study was sponspored by the food industry given that participant found the product favorable is

P(F|A) = [P(A|F)P(F) ] / P(A) = [0.62*0.60] / 0.56 = 0.6643

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By the law of total probability, the probability that a participants found the product unfavorable is

P(B) = P(B|F)P(F) + P(B|N)P(N) = 0.13 * 0.60 + 0.40 * 0.40 = 0.238

By the Baye's theorem, the probability that the study had no industry funding given that participant found the product unfavorable is

P(N|B) = [P(B|N)P(N) ] / P(B) = [0.40*0.40] / 0.238 = 0.6723

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