Question

To save money, lab usually merges blood samples for test. Samples from multiple people will be tested only once, if the result is negative (i.e. no virus, every criterion is in the normal range, etc.), then all these people tested are healthy. If the result is positive (at least one person from this batch whose blood sample is abnormal), then these samples are tested one-by-one. Suppose all samples are taken independently. If the probability that for a person to get a disease is 0.005

1. What is the probability that the batch test with 30 people’s sample is positive?

2. What is the expected times of test with this method if 5 samples are merged? Does it save money?

Answer 1

1)probability that the batch test with 30 people’s sample is positive

=P(at least one person have disease)=1-P(none have dieases)=1-(1-0.005)³⁰ =0.1396

2)

here probability that batch is negative means one test is done =(1-0.005)⁵ =0.975249

probability that batch is positive means 6 test needs to be done =1-0.975249=0.024751

hence expected test E(X)=1*0.975249+6*0.024751=1.123756

as expected number of test from this method is less than 5 whcih will happen if tests are done seperately theefore it saves money,.