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Embarrassing Questions, German Tanks and Estimations



Q: You are conducting a survey and want to ask an embarrassing yes/no question to subjects. The subjects wouldn't answer that embarrassing question honestly unless they are guaranteed complete anonymity. How would you conduct the survey?

Machine Learning: The Art and Science of Algorithms that Make Sense of Data

A: One way to do this is to assign a fair coin to the subject and ask them to toss it in private. If it came out heads then answer the question truthfully else toss the coin a second time and record the result (heads = yes, tails = no). With some simple algebra you can estimate the proportion of users who have answered the question with a yes.

Assume total population surveyed is \(X\). Let \(Y\) subjects have answered with a "yes". Let \(p\) be the sort after proportion. The tree diagram below shows the user flow.



The total expected number of "yes" responses can be estimated as
$$
\frac{pX}{2} + \frac{X}{4} = Y
$$
which …