S Randomised response. In a survey on a delicate topic (‘Do you take hard drugs?’) it is difficult to protect the privacy of the people questioned and at the same time to get reliable answers. That is why the following ‘unrelated question method’ was suggested. A deck of cards is prepared such that half of the cards contain the delicate question A and the other half a harmless question B, which is unrelated to question A (‘Did you go to the cinema last week?’).

The interviewer asks the candidate to shuffle the cards, then to choose a card without showing it to anyone, and to answer the question found on this card. The group of people questioned contains a known proportion pB of people affirming question B (cinema-goers). Let # D pA be the unknown probability that the sensitive question A is answered positively. Suppose n people are questioned independently. Specify the statistical model, find a best estimator of #, and determine its variance.