2.6.10 problem 69

(a)

$$y=\mathrm{dp}+(1-d)(1-p)$$

(b)

The worst choice for $p$ is $\frac{1}{2}$, because then the fraction of "yes" responses is $\frac{1}{2}$ irrespective of the fraction of drug users. In other words, the number of "yes" responses tells us nothing.

(c)

We can extend the result from part $a$.

A drug user says "yes" either if they get a "Have you used drugs" slip, or if they get a "I was born in winter" slip and they are, in fact, born in winter.

A person who has not used drugs says "yes" only in the case that they get a "I was born in winter" slip and they were, in fact, born in winter.

$$y=d(p+\frac{1}{4}(1-p\left)\right)+(1-d)(1-p)\frac{1}{4}$$

Thus,

$$d=\frac{4y+p-1}{2p}$$