A researcher wants to estimate the percentage of people in somepopulation who have used illegal drugs, by conducting a survey.Concerned thata lot of people would lie when asked a sensitive question like Have you ever usedillegal drugs?, the researcher uses a method known asrandomized response.A hatis lled with slips of paper, each of which says either I have used illegal drugs orI have not used illegal drugs.Letpbe the proportion of slips of paper that sayI have used illegal drugs (pis chosen by the researcher in advance).Each participant chooses a random slip of paper from the hat and answers (truth-fully) yes or no to whether the statement on that slip is true.The slip is thenreturned to the hat.The researcher does not know which type of slip the partici-pant had.Letybe the probability that a participant will say yes, anddbe theprobability that a participant has used illegal drugs.

(a) Findy, in terms ofdandp.

(b) What would be the worst possible choice ofpthat the researcher could make indesigning the survey?Explain.

(c) Now consider the following alternative system.Suppose that proportionpof theslips of paper say I have used illegal drugs, but that now the remaining 1psayI was born in winter rather than I have not used illegal drugs.Assume that1/4 of people are born in winter, and that a persons season of birth is independentof whether they have used illegal drugs.Findd, in terms ofyandp.