Context : To get answers to sensitive questions, we sometimes use a method called the randomized response technique. Suppose, for instance, that we want to determine what percentage of the students at a large university take ketamine. We construct 20 flash cards, write ‘I take ketamine at least once a week’ on 12 of the cards (where 12 is an arbitrary choice) and ‘I do not take ketamine at least once a week’ on the others. Then we let each student (in the sample interviews) select one of the 20 cards at random, and response yes or no without divulging the question.
Establish a relationship between P(Y), the probability that a student will give a yes response, and P(K), the probability that a student randomly selected at the university takes ketamine at least once a week.
I received the following question as an undergraduate Statistics student, and I am confused about the whole idea of the "randomised response technique". The process describes a student choosing the cards and they say yes or no, but how can we determine the percentage of students that may actually take ketamine? If I say that the probability a student gives a yes response is 0.5, this would imply that half the population in the university takes ketamine, although this would be incorrect. Could someone please explain how do I derive the probability more accurately?