Maddenation
Asking an embarrassing question
I have been looking through the book on probability puzzles the Patrick and Karina gave me. This is definitely not a book for everyone. But for me, it’s perfect. All kinds of equations and formulas and really tough problems. Not to mention that the author, Paul Nahin, is probably someone we can ask to resolve our long-standing Survivor voting question. (Basically, why there weren’t more ties.)
The first, and probably easiest, problem in the book concerns a psychologist trying to gather data on the percentage of people performing some embarrassing private act. (He didn’t say what.) Of course, you can’t expect people to answer honestly unless their identity is protected. The question is whether one can devise a way to ask the embarrassing question one person at a time and still protect their anonymity.
Alas, you can, and Nahin gives away the answer in his question. He read about it in a medical journal. Here’s how it works. You give the subject a coin and send them into a private room with the following instructions. Flip the coin before answering the question. If the coin comes up tails, answer the embarrassing question honestly (yes or no). If the coin comes up heads, flip it again and answer this second yes/no question. “Did the coin come up heads again?”
The person comes out of the room and hands you a piece of paper with “yes” or “no” on it, but at that point you don’t know which question they were answering. After 10,000 people have been asked, you tally the answers and find that there are 6230 yeses and 3770 noes. How can these data be used to determine the fraction of people in the survey who practiced the private act?
Dad • Puzzles • 12/30/04 • 5 comments
Comments
Patrick • 01/02/05 • 6:10 PM:This assumes that the people who flip tails are 1) really going to answer the question honestly, and 2) are really going to tell you their coin came up tails. Don’t you think? Unless there’s something I’m not seeing. I think that your percentage is correct (given the assumptions I mentioned above). If you want to be more correct about it, you can just halve both numbers, figuring that half the people will have flipped heads, and of those, half will flip heads again (yielding a false “yes”) and the other half will flip tails the second time (yielding a false “no”).
But, again, I don’t really see how this is better than simply asking people to be honest, and I think it will suffer from the people who really don’t want to answer the question (likely “yes” answerers) who lie about their coin flip in the first place… Unless I’m missing something.
Dad • 01/02/05 • 7:24 PM:Yes, it assumes the people who flip tails will answer honestly, but nobody has to tell you anything about coin flips. They simply hand you the paper with Yes of no on it. Maybe what you’re missing is that only one question is answered, either the real question (if the first flip comes up tails), or the result of the second flip (if it comes up heads). Yes, these studies can always be sabotaged by someone willing to lie, but why would they lie? The process protects their “private” information because your yes or no could always be interpreted as the answer to the second question.
Your idea about halving the numbers is a good idea, but not correct.
Patrick • 01/02/05 • 10:06 PM:Ah. So you really just toss 2,500 yeses and 2,500 noes, and then do your results out of 5,000 respondents, right? Sorry for my wrongness.
Patrick • 01/02/05 • 11:58 PM:Thinking more about this, I wonder if it might lead to a different problem: lots of people feeling they have nothing to lose by simply answering honestly. So instead of tossing the coin, they just answer honestly. And maybe more “no” people are answering honestly than “yes” people, so that skews your results…
Dad • 01/03/05 • 11:20 AM:You are right about tossing out 2500 y and 2500 n answers. Your second comment I don’t agree with. The “nothing to lose” people would not be answering “honestly” if they chose not to follow the directions. The few that do decide to mess with the results would probably fall within the error band. Even with 10000 fair coin flips, you can’t expect to get exactly 5000 heads. Which raises the question, “Within what range would you expect your results to be?” That is, what is the standard deviation in the number of heads thrown in 10000 coins tosses?
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