Bayes Theorem probability question?

Question

Suppose we want to estimate the fraction of SUNY students who have shoplifted. Instead of asking the incriminating question question directly, we ask each student being surveyed to toss a coin in private. If the coin lands "heads," they are to answer the question, "Have you shoplifted?" truthfully. If the coin shows "tails," they are told to always answer "yes" no matter what. If the true fraction of students who have shoplifted is in fact 0.15, and if the participants in the survey answer accurately and honestly, what is the conditional probability that a randomly chosen student who answers "yes" to the survey has in fact shoplifted?

Thanks mucho, in advance, for your expertise!

-j. · Accepted Answer

There are two groups:  truth-tellers and yes-sayers.  Assuming the coin is fair, approximately half of the students will be in each group.

0.15 of the truth-tellers will answer yes.  That's 0.15 * 0.5 of the population, or 0.075.
1.00 of the yes-sayers will answer yes.  That's 1.00 * 0.5 of the population, or 0.5.

So, probabilistically, 0.575 of the population will answer yes.  Of those, only 0.15 (the "true fraction") have actually shoplifted.  So
0.15 / 0.575 = 0.26 (rounded)

Philip · Answer

This system was devised to counter the fact that shoplifters wouldn't tell the truth.
50% of the participants will answer yes due to the fact that they tossed a tails.
Of the remaining 50%, 0.15 = 7.5% will answer yes if everyone tells the truth. They are more likely to tell the truth being hidden among another 50% of yes answers. Rational being, nobody knows if they said yes because they shoplifted or because they tossed a tails.
So the conditional probability a randomly chosen student who answer yes is 7.5%*2 out of 57.5% = .260

Bayes Theorem probability question?

2 Answers