# Warner’s randomised response model Question: What percentage of students have cheated during an exam in college? Problem: Students may be too embarrassed to answer truthfully Randomisation to the rescue: We do a survey that first instructs students to toss a coin twice. If the student gets 'tails' on the first toss, then the student has to answer question 1, otherwise the student answers question 2: 1. Have you every cheated on an exam in college? 2. Did you get tails on the second toss? The answer will be partly random. We don't know whether a yes answer is due to the student cheating or to getting tails on the second toss. This should put the student at ease to answer truthfully. Key point: while we don't know what an individual yes means, we can estimate the proportion of cheaters using the answers collectively: P(yes) = P(yes and Q1) + P(yes and Q2) P(yes) = P(yes | Q1) P(Q1) + P(yes | Q2) P(Q2) Solve for $P(\text{yes } | Q1) = \frac{P(\text{yes}) - P(\text{yes } | Q2) P(Q2)) }{P(Q1)}$ We need to estimate P(yes) from data. The rest is just plug and play