Question 1168489
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