Chapter 4, Problem 4.65P

Each of 500 soldiers in an army company independently has a certain disease with probability $\frac{1}{{10}^3}$. This disease will show up in a blood test, and to facilitate matters, blood samples from all 500 soldiers are pooled and tested, a. What is the (approximate) probability that the blood test will be positive (that is, at least one person has the disease)? Suppose now that the blood test yields a positive result.

b. What is the probability, under this circumstance, that more than one person has the disease? Now, suppose one of the 500 people is Jones, who knows that he has the disease.

c. What does Jones think is the probability that more than one person has the disease? Because the pooled test was positive, the authorities have decided to test each individual separately. The first i -1 of these tests were negative, and the with one—which was on Jones—was positive.

d. Given the preceding scenario, what is the probability, as a function of i, that any of the remaining people have the disease?