Suppose that about 20% of students in school have used marijuana in the last yea

Question

Suppose that about 20% of students in school have used marijuana in the last year. In order to preserve anonymity during surveys, a technique called randomized response is often used. A student is approached and asked to flip a coin but not show it to the interviewer. If the coin is heads, the student answers the question: “Was your mother born in January to June (inclusive)?” If the coin is tails, the student answers the question: “Have you used marijuana in the last year?” Because the interviewer does not know the outcome of the coin flip, the students' responses are confidential. What is the probability then that if a yes is received, the student used marijuana in the last year?

Explanation / Answer

P(the student used marijuana in the last year | a yes received ) = P(Yes is received and student also used marijuana)/P(a yes is received)

P(Yes is received and student also used marijuana)= P(tail)*P(a random student uses marijuana)= 0.5*0.2=0.1

P(a yes is received) = P(tail)*P(a random student uses marijuana)+P(head)*P(mother is born in January to June)

= 0.5*0.2 + 0.5*0.5 = 0.10 + 0.25 = 0.35

Required probabilty = 0.10/0.35 = 0.2857

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