In a college course, three different professors are teaching their classes: Professor A, Professor B, and Professor C. Students are enrolling in one of these courses, and the following probabilities are known: • The probability that a randomly selected student is in Professor A's class is 0.4. • The probability that a randomly selected student is in Professor B's class is 0.3. • The probability that a randomly selected student is in Professor C's class is 0.3. Students have also reported their satisfaction with the course, particularly in terms of teaching quality: • 80% of students in Professor A's class report excellent teaching quality. • 60% of students in Professor B's class report excellent teaching quality. • 70% of students in Professor C's class report excellent teaching quality. Now, if a student is satisfied with their course's teaching quality, what is the probability that they are enrolled in Professor A's class?
Question
In a college course, three different professors are teaching their classes: Professor A, Professor B, and Professor C. Students are enrolling in one of these courses, and the following probabilities are known: • The probability that a randomly selected student is in Professor A's class is 0.4. • The probability that a randomly selected student is in Professor B's class is 0.3. • The probability that a randomly selected student is in Professor C's class is 0.3. Students have also reported their satisfaction with the course, particularly in terms of teaching quality: • 80% of students in Professor A's class report excellent teaching quality. • 60% of students in Professor B's class report excellent teaching quality. • 70% of students in Professor C's class report excellent teaching quality. Now, if a student is satisfied with their course's teaching quality, what is the probability that they are enrolled in Professor A's class?
Solution
To solve this problem, we need to use Bayes' theorem, which is a way to find a probability when we know certain other probabilities. The formula for Bayes' theorem is:
P(A|B) = [P(B|A) * P(A)] / P(B)
In this case, we want to find the probability that a student is in Professor A's class given that they are satisfied with the course's teaching quality. So, we can assign the events as follows:
Event A: The student is in Professor A's class. Event B: The student is satisfied with the teaching quality.
We know the following probabilities:
P(A) = 0.4 (probability that a student is in Professor A's class) P(B|A) = 0.8 (probability that a student is satisfied with the teaching quality given that they are in Professor A's class)
We also need to find P(B), the total probability that a student is satisfied with the teaching quality. We can find this by considering all the ways a student can be satisfied, which is by being in any of the three classes. So,
P(B) = P(B and A) + P(B and not A) = P(B|A)P(A) + P(B|B)P(B) + P(B|C)P(C) = (0.8)(0.4) + (0.6)(0.3) + (0.7)(0.3) = 0.32 + 0.18 + 0.21 = 0.71
Now we can substitute these values into Bayes' theorem:
P(A|B) = [P(B|A) * P(A)] / P(B) = [0.8 * 0.4] / 0.71 = 0.32 / 0.71 = 0.4507
So, the probability that a student is in Professor A's class given that they are satisfied with the course's teaching quality is approximately 0.45, or 45%.
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