# How to Ask an Embarrassing Question

I despise telemarketers and hate filling questionnaires. Usually there are too many questions, most either irrelevant (in my view) and rather personal. I may not mind answering a single question, though, especially if that requires a simple Yes/No answer. Provided, of course, it does not touch on a private matter.

And what if it does?

What if a question XXX is of the potentially embarrassing sort that most people would not care to answer unless assured of complete anonymity? But how complete anonymity may be assured?

Here's how: let each person from a random population toss a (fair) coin. If it shows tails ask the person to truthfully answer the XXX question, Yes or No. If the coin shows heads, instruct the person to record the result of the second toss, with Yes for, say, heads and No for the tails. Ask the person to report the final Yes/No result.

As the Yes/No answers are collected there is no way to tell whether these relate to the XXX question or to the second toss of a coin. However, there is a way to estimate the proportion of the respondents that answered the XXX question one way or the other. How is this done?

### References

1. E. B. Burger, M. Starbird, The Heart of Mathematics, Key College Publishing, 2000, Ch 7.6
2. P. Nahin, Duelling Idiots and Other Probability Puzzlers, Princeton University Press, 2000, pp 15-16