Question 1168489: 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, 15 % of the participants found the products unfavorable, 23 % were neutral, and 62 % found the products favorable. Of those that had no industry funding, 38 % found the products unfavorable, 15 % were neutral, and 47 % found the products favorable.
a)What is the probability that a participant selected at random found the products favorable?
b)If a randomly selected participant found the product favorable, what is the probability that the study was sponsored by the food industry?
c)If a randomly selected participant found the product unfavorable, what is the probability that the study had no industry funding?
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step by step.
**1. Organize the Data**
* **Industry Sponsored (IS):** 60 studies
* Unfavorable (U_IS): 15% of 60 = 0.15 * 60 = 9 studies
* Neutral (N_IS): 23% of 60 = 0.23 * 60 = 13.8 studies (round to 14)
* Favorable (F_IS): 62% of 60 = 0.62 * 60 = 37.2 studies (round to 37)
* **No Industry Funding (NI):** 40 studies
* Unfavorable (U_NI): 38% of 40 = 0.38 * 40 = 15.2 studies (round to 15)
* Neutral (N_NI): 15% of 40 = 0.15 * 40 = 6 studies
* Favorable (F_NI): 47% of 40 = 0.47 * 40 = 18.8 studies (round to 19)
* **Total Studies:** 60 + 40 = 100 studies
* **Total Favorable:** 37 + 19 = 56 studies
* **Total Unfavorable:** 9 + 15 = 24 studies
* **Total Neutral:** 14 + 6 = 20 studies
**a) What is the probability that a participant selected at random found the products favorable?**
* P(Favorable) = (Number of Favorable Studies) / (Total Studies)
* P(Favorable) = 56 / 100 = 0.56
**b) If a randomly selected participant found the product favorable, what is the probability that the study was sponsored by the food industry?**
* We need to find P(IS | Favorable), which is the probability of the study being industry-sponsored given that the participant found the product favorable.
* Using Bayes' Theorem:
* P(IS | Favorable) = [P(Favorable | IS) * P(IS)] / P(Favorable)
* P(Favorable | IS) = 37 / 60 ≈ 0.6167
* P(IS) = 60 / 100 = 0.6
* P(Favorable) = 0.56
* P(IS | Favorable) = (0.6167 * 0.6) / 0.56
* P(IS | Favorable) = 0.37 / 0.56
* P(IS | Favorable) ≈ 0.6607
**c) If a randomly selected participant found the product unfavorable, what is the probability that the study had no industry funding?**
* We need to find P(NI | Unfavorable), which is the probability of the study having no industry funding given that the participant found the product unfavorable.
* Using Bayes' Theorem:
* P(NI | Unfavorable) = [P(Unfavorable | NI) * P(NI)] / P(Unfavorable)
* P(Unfavorable | NI) = 15 / 40 = 0.375
* P(NI) = 40 / 100 = 0.4
* P(Unfavorable) = 24 / 100 = 0.24
* P(NI | Unfavorable) = (0.375 * 0.4) / 0.24
* P(NI | Unfavorable) = 0.15 / 0.24
* P(NI | Unfavorable) = 0.625
**Answers:**
a) 0.56
b) Approximately 0.6607
c) 0.625
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