At the beginning of the semester, a professor tells students that if they study for the tests, then there is a 55% chance they will get a B or higher on the tests. If they do not study, there is a 20% chance that they will get a B or higher on the tests. The professor knows from prior surveys that 60% of students study for the tests. The probabilities are displayed in the tree diagram.
A tree diagram. Random student to studies for test is 0.6, and to does not study for test is 0.4. Studies for test to Gets B or higher is 0.55; does not get B or higher is 0.45. Does not study for test to Gets B or higher is 0.20; does not get B or higher is 0.80.
The professor informs the class that there will be a test next week. What is the probability that a randomly selected student studied for the test if they pass it with a B or higher?
0.20
0.55
0.60
0.80
To find the probability that a student studied for the test given that they passed it with a B or higher, we need to use Bayes' theorem.
Let A be the event that a randomly selected student studied for the test and B be the event that the student passed the test with a B or higher. We want to find P(A|B), the probability that a student studied for the test given that they passed it with a B or higher.
Using Bayes' theorem, we have:
P(A|B) = (P(B|A) * P(A)) / P(B)
P(B|A) = 0.55 (given)
P(A) = 0.60 (given)
P(B) = (P(B|A) * P(A)) + (P(B|not A) * P(not A))
= 0.55 * 0.60 + 0.20 * 0.40
= 0.33 + 0.08
= 0.41
Plugging in the values, we get:
P(A|B) = (0.55 * 0.60) / 0.41
= 0.33 / 0.41
≈ 0.8049
Therefore, the probability that a randomly selected student studied for the test, given that they passed it with a B or higher, is approximately 0.8049. So the closest option is 0.80.