In a recent survey in a Statistics class, it was determined that only 60% of the students

attend class on Thursday. From past data it was noted that 98% of those who went to class
on Thursday pass the course, while only 20% of those who did not go to class on Thursday
passed the course.
(a) What percentage of students is expected to pass the course?
(b) Given that a student passes the course, what is the probability that he/she attended classes
on Thursday.

To solve this problem, we can use conditional probability.

(a) To find the percentage of students expected to pass the course, we need to consider both the students who attended class on Thursday and those who did not.

Let's assume there are 100 students in total.

Out of these 100 students, 60% attended class on Thursday, which is 60 students. Since 98% of these students passed the course, we can calculate the number of students who passed the course:

Number of students who attended class on Thursday and passed = (60 * 98%) = 58.8 ≈ 59 students.

Now, let's consider the students who did not attend class on Thursday. From the given information, we know that 20% of these students passed the course. Therefore, we can calculate the number of students who did not attend class on Thursday but still passed the course:

Number of students who did not attend class on Thursday and passed = (40 * 20%) = 8 students.

Now, to calculate the percentage of students expected to pass the course, we add the number of students who passed from both groups and divide it by the total number of students:

Percentage of students expected to pass the course = (59 + 8) / 100 * 100% = 67%.

Therefore, the expected percentage of students who will pass the course is 67%.

(b) To find the probability that a student attended class on Thursday given that they passed the course, we can use Bayes' theorem.

Let's calculate each component of Bayes' theorem:

P(A) = Probability that a student attended class on Thursday = 60%

P(B) = Probability that a student passed the course = 67%

P(A|B) = Probability that a student attended class on Thursday given that they passed the course

We know that P(A|B) = P(A ∩ B) / P(B), where P(A ∩ B) is the probability that a student both attended class on Thursday and passed the course.

From the previous calculations, we know that the number of students who attended class on Thursday and passed = 59 and the total number of students who passed the course = 67.

Therefore,

P(A|B) = (Number of students who attended class on Thursday and passed) / (Total number of students who passed the course)

P(A|B) = 59 / 67 ≈ 0.88.

So, the probability that a student attended class on Thursday given that they passed the course is approximately 0.88 or 88%.

To answer these questions, we need to use conditional probability. Let's tackle them step by step:

(a) To determine the percentage of students expected to pass the course, we can use the law of total probability. We'll calculate the weighted average of the percentage of students passing the course for both those who attended class on Thursday and those who didn't.

Let's denote:
P(P) = Probability of passing the course
P(T) = Probability of attending class on Thursday
P(P|T) = Probability of passing the course given attendance on Thursday
P(P|T') = Probability of passing the course given no attendance on Thursday

From the information given:

P(T) = 60% (proportion of students who attend class on Thursday) = 0.6
P(T') = 1 - P(T) = 0.4

P(P|T) = 98% (those who attended class on Thursday pass the course) = 0.98
P(P|T') = 20% (those who didn't attend class on Thursday pass the course) = 0.2

Now, applying the law of total probability:

P(P) = P(T) * P(P|T) + P(T') * P(P|T')
P(P) = 0.6 * 0.98 + 0.4 * 0.2

Calculating this expression, we find:
P(P) = 0.588 + 0.08
P(P) = 0.668

Therefore, the percentage of students expected to pass the course is 66.8%.

(b) To determine the probability that a student attended class on Thursday given that they passed the course, we use conditional probability.

Let's denote:
P(T|P) = Probability of attending class on Thursday given passing the course

To calculate this conditional probability, we'll use Bayes' theorem:

P(T|P) = (P(P|T) * P(T)) / P(P)
P(T|P) = (0.98 * 0.6) / 0.668

Calculating this expression, we find:
P(T|P) = 0.588 / 0.668
P(T|P) ≈ 0.879

Therefore, the probability that a student attended class on Thursday given that they passed the course is approximately 87.9%.

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