Testing for Groate\'s Disease is very expensive, and positive tests are quite ra

ID: 2506209 • Letter: T

Question

Testing for Groate's Disease is very expensive, and positive tests are quite rare. To cut costs, a

hospital considers the following procedure: they pool blood samples from twenty patients (while preserving

the individual samples), and test the pooled blood. If the test on the pooled blood sample is negative, all 20

patients test negative. If the pooled test is positive, the samples are then tested individually to determine

where the positive blood came from. Suppose that the probability any one individual's blood is positive is. 018, and that the

samples are independent across patients.

a. What is the probability that a pool of 20 individual samples will test negative?

b. What is the probability that a pool of 20 individual samples will test positive?

c. Let N represent the number of tests required to determine positive/negative status of a group of 20

patients, when following the pooled sample procedure described above. What is the expected value of N?

d. Dr. House suggests that while the pooled procedure is superior to testing each sample individually, the

hospital can save even more money by pooling samples from X individuals, where X does not equal 20. Show that he is 6

correct.

Probability of an individual testing positive = .018

Probability of an individual testing negavie = .982

Probability that a pool of 20 individuals samples will test negative = 20C20 * .982^20 * .018^0 = .982^20 = .6953921051

Probability that a pool of 20 individual samples will test positive = 1 - .6953921051 = .3046078949

Number of tests required if pool is negative = 1

Number of test required if pool is positive = 21 (1 pooled and 20 individual)

Expected number of tests = 1(.6953921051) + 21(.3046078949) = 7.092157899

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