Question

Scientific research on popular beverages consisted of 65 studies that were fully sponsored by the food industry, and 35 studies that were conducted with no corporate ties. Of those that were fully sponsored by the food industry, 10 % of the participants found the products unfavorable, 25 % were neutral, and 65 % found the products favorable. Of those that had no industry funding, 39 % found the products unfavorable, 13 % were neutral, and 48 % 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?

Answer 1

From the given percentages, we find the actual numbers for each cell as below

	Sponsored	No corporates	Total
Unfavourable	6.5	13.65	20.15
Neutral	16.25	4.55	20.8
Favourable	42.25	16.8	59.05
Total	65	35	100

1) P(participant found product favourable)      
= Total participants who found Favourable / Total participants      
= 59.05/100      
= 0.5905      

P(participant found product favourable) = 0.5905
       
2) Let A : event that participant found product favourable      
Let B : event that the study was sponsored by food industry      
To find P(B | A)      
P(B | A) = P(B AND A) / P(A)      
P(B AND A) = 42.25/100      
             = 0.4225      
P(A) = 0.5905  (As found in (1))    
P(B|A) = 0.4225/0.5905      
                = 0.7155      
P(study was sponsored by the food industry given participant found product favourable) = 0.7155      
       
3) Let A : event that participant found product unfavourable      
Let B : event that the study had no industry funding      
To find P(B | A)      
P(B | A) = P(B AND A) / P(A)      
P(B AND A) = 13.65/100      
             = 0.1365      
P(A) = 20.15/100 = 0.2015      
P(B|A) = 0.1365/0.2015      
                = 0.6774      
P(study had no industry funding given participant found product unfavourable) = 0.6774