7. [10 points] In a particular college statistics course, twenty percent of the

Question

7. [10 points] In a particular college statistics course, twenty percent of the students had previously taken a statistics course during high school while the others had not. Among those students who had taken a high school course, twenty-five percent received an A in their college course; while only fifteen percent of those students with no prior exposure received an A in the college course Suppose that one student from the college course is selected at random. (a) Find the probability that they had taken statistics during high school and receive an A in their college course. [3 points] (b) Find the probability that they receive an A in the college course. [3 points] (c) Suppose that you overhear the student boasting about receiving an A in their college statistics course. Find the probability that they had not taken a statistics course during high school. 4 points]

Explanation / Answer

Let A denote the event that a student had taken statistics in high school and B that the student received A in college.

Given P(A) = 0.24 P(B|A) = 0.25 P(B|Ac) = 0.15

(a) Using Baye's theorem we have

P(B|A) = P(A B) / P(A)

=> P(A B) = P(A) P(B|A)

=> P(A B) = 0.24 * 0.25

= 0.06

(b) P(Ac) = 1 - P(A)

= 1 - 0.24

= 0.76

P(Ac B) = P(Ac) P(B|Ac)

=> P(A B) = 0.76 * 0.15

= 0.114

P(B) = P(A B) + P(Ac B)

= 0.06 + 0.114

= 0.174

(c) P(Ac|B) = P(Ac B) / P(B)

= 0.114 / 0.174

= 0.655

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