Mathematical Solutions for Maintaining Privacy

The idea of obtaining demographic and other information without compromising personal privacy has been around for a long time. For a different sort of illustration, let's assume we have a large group of people and we want to discover what percentage of them have done something, say X, that they'll probably be embarrassed to admit. Assume also that there is a legitimate reason, say medical or otherwise, for our wanting to know what percentage of people have X-ed. What can we do?

Again we use a randomizing device and ask each person in the group to flip a coin and keep the result secret. If the coin lands heads, the person is instructed to answer honestly: Has he or she ever X-ed — Yes or No? If the coin lands tails, the person is instructed to simply answer Yes. Thus a Yes response could mean one of two things, one quite innocuous (the coin landing tails), the other potentially embarrassing (X-ing). Since no one can know what the Yes means, presumably people will be honest.

For illustration, let's say that 560 of 1000 responses are Yes. What does this indicate about the percentage of people who have X-ed? Approximately 500 of the 1000 people will answer Yes simply because their coin landed tails. We'll ignore them and focus only on the 500 people whose coin landed heads and who therefore replied to the question honestly. Of these 500 approximately 60 answered Yes. Thus 12 percent (60/500) is the estimate for the percentage of people who have X-ed. There are many refinements of this method that can be used to learn more detail, such as how many times people have X-ed.

Until such techniques become widespread, lying is a reasonable strategy when confronted with overly probing questions. Time for my Hawaiian ice fishing lesson.

Professor of mathematics at Temple University and adjunct professor of journalism at Columbia University, John Allen Paulos is the author of several best-selling books, including Innumeracy, and the forthcoming A Mathematician Plays the Market, which will be published in the spring. His Who’s Counting? column on appears the first weekend of every month.

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