# Simple cross-tabulated data and probability

I saw this question which I found confusing: Suppose 400 people are surveyed. For each person, they flip a coin and if it lands heads they are asked,

A: Are you a student?

B: Do you shoplift?

230 of the 400 surveyed answered yes. Of the students, estimate the proportion who shoplift.

Personally I would attempt to solve this with conditional probabilities, but it seems to me there is inadequate information for that. Here's my attempt.

$$\frac{230}{400} = P(Yes)$$

$$P(Shoplift|Student) = P(Yes|Tails)$$

$$P(Yes) = P(Yes | Tails)P(Tails) + P(Yes|Heads)P(Heads)$$

$$P(Tails)=P(Heads) = .5$$

Other than this I don't think any information is given and useful, and this amount of information seems inadequate to solve for $P(Shoplift|Student)$. The most you can infer is

$$\frac{230}{400} = (.5)(P(Yes|Tails) + P(Yes|Heads)) \Rightarrow$$

$$P(Yes|Tails) = \frac{23}{20} - P(Yes|Heads)$$