I was given this question in college. We were not shown anything like this before. Nobody in the class or in the college maths centre was able to solve it so far. Would anybody have an idea?
It was anticipated that it would not be possible to assess the extent of tax invasion by direct questioning, so it was decided to use the randomised response technique. Each respondent was given a card with the following two questions:
Was your mother born in April? Have you ever evaded tax?
Then the respondent was asked to toss a coin and answer the first question if a head turned up or answer the second question if a tail turned up. The interviewer did not know which question was answered. Out of 1000 people interviewed, 200 answered yes. Estimate the proportion of tax evaders to two decimal places.
The basic idea is that we can define two probabilities, one for each of the questions. Let $p_1$ represent the probability that a randomly selected respondent's mother was born in April. Let $p_2$ represent the probability that a randomly selected respondent has ever evaded tax.
Now, assuming the coin is fair, half of the $1000$ respondents answered the first question, of which $500p_1$ answered "Yes," and half answered the second question, of which $500p_2$ answered "Yes." So the total number of respondents answering "Yes"--we are told $200$--should be the sum $$200 = 500(p_1 + p_2).$$ Therefore, if you impose an assumption on the value of $p_1$, then $p_2$ can be estimated.
This of course gives you a point estimate of the probability. To properly characterize the variance of your estimate, you would need to do more work.