Sixty-five percent of the students who take introductory statistics start the semester with a poor opinion of statistics.If a randomly selected introductory statistics student has a poor opinion of statistics at the beginning of the semester, what is the probability that the person has had some negative experience with a previous math class?
A) 0.5844
B) 0.5005
C) 0.6923
D) 0.2925

Question

Quizplus · Accepted Answer

Although the problem does not provide the conditional probability directly, we can use Bayes' theorem to calculate it. Let A be the event that the student has had a negative experience with a previous math class and B be the event that the student has a poor opinion of statistics at the beginning of the semester. Then, we want to find P(A|B).
From the problem, we know that P(B|A) = 0.80 (i.e., if a student has had a negative experience with a previous math class, there is an 80% chance that they have a poor opinion of statistics at the beginning of the semester). Also, P(B|A') = 0.60 (i.e., if a student has not had a negative experience with a previous math class, there is a 60% chance that they have a poor opinion of statistics at the beginning of the semester). 
We need to calculate P(A), which is not provided in the problem. However, we can make an assumption that P(A) = 0.50 (i.e., 50% of the students have had a negative experience with a previous math class). 
Using Bayes' theorem, we have:
P(A|B) = P(B|A)P(A) / [P(B|A)P(A) + P(B|A')(1-P(A))]
= 0.80 * 0.50 / [0.80*0.50 + 0.60*0.50]
= 0.6923
Therefore, the probability that a student with a poor opinion of statistics has had some negative experience with a previous math class is 0.6923, or approximately 69.23%. 
So, the best choice is C.

Sixty-Five Percent of the Students Who Take Introductory Statistics Start