In a certain school, 17 percent of the students are enrolled in a psychology course, 28 percent are enrolledin a foreign language course, and 32 percent are enrolled in either a psychology course or a foreign languagecourse or both. What is the probability that a student chosen at random from this school will be enrolled inboth a foreign language course and a psychology course?(A) 0.45(B) 0.32(C) 0.20(D) 0.13(E) 0.05
Question
In a certain school, 17 percent of the students are enrolled in a psychology course, 28 percent are enrolledin a foreign language course, and 32 percent are enrolled in either a psychology course or a foreign languagecourse or both. What is the probability that a student chosen at random from this school will be enrolled inboth a foreign language course and a psychology course?(A) 0.45(B) 0.32(C) 0.20(D) 0.13(E) 0.05
Solution
To solve this problem, we can use the principle of inclusion and exclusion. This principle states that for any two sets, the probability of either event occurring is the sum of the probabilities of each event occurring separately, minus the probability of both events occurring.
Here, the two events are "student is enrolled in a psychology course" and "student is enrolled in a foreign language course".
The problem states that:
- The probability of a student being enrolled in a psychology course (P(P)) is 17 percent or 0.17.
- The probability of a student being enrolled in a foreign language course (P(F)) is 28 percent or 0.28.
- The probability of a student being enrolled in either a psychology course or a foreign language course or both (P(P U F)) is 32 percent or 0.32.
We want to find the probability of a student being enrolled in both a psychology course and a foreign language course (P(P ∩ F)).
According to the principle of inclusion and exclusion, P(P U F) = P(P) + P(F) - P(P ∩ F).
We can rearrange this formula to solve for P(P ∩ F):
P(P ∩ F) = P(P) + P(F) - P(P U F) = 0.17 + 0.28 - 0.32 = 0.13.
So, the probability that a student chosen at random from this school will be enrolled in both a foreign language course and a psychology course is 0.13 or 13 percent.
Therefore, the answer is (D) 0.13.
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