A way to make screening tests more economical is to pool samples. Suppose that we would like to screen for a rare blood condition that affects 1 out of every 1000 people within a population. In order to screen for this condition in a small town of 1500 people we have devised the following screening method:
1) Pool samples into groups of 500.
2) Test each of the pooled samples. The test will give a positive if one or more of the subjects included in the pooled sample have the condition.
3) If the pooled sample is positive then individually test each of the 500 samples in the pool.
a) What is the probability that we obtain a positive test result for a randomly chosen pooled sample?
b) What is the probability that more than one of the subjects in a given pool is positive given that the pool tested positive?
c) Define a random variable X. Let X equal the number of samples tested using the above screening method for the small town. What is the probability distribution of X?
d) What is the expected value of X?