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. (5 points 2.5 for each)

a) What is the probability that the batch test with 30 people’s sample is positive?

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

Answer 1

Hello Sir/ Mam

Let A be the random variable that represent the number of people having disease.

P(Person got a disease) = 0.005

(a) A batch of 30 people was tested. The sample will come positive even if one person has got infection.

Hence,

(b)

If the test is negative, there will be just 1 test, however is the test is positive, tere will be 1+5=6 tests. Hence,

A	P(A)	Number of times test have to be performed
0	0.9752487531219	1	0.975248753122
1	0.0245037375156	6	0.147022425094
2	0.0002462687188	6	0.001477612313
3	0.0000012375313	6	0.000007425188
4	0.0000000031094	6	0.000000018656
5	0.0000000000031	6	0.000000000019
Total			1.123756234391

The expected number of tests will be 1.1238 as compared to 5 tests if they are tested individually(all 5 one by one). Hence, this method saves money.

I hope this solves your doubt.

Do give a thumbs up if you find this helpful.