A medical clinic tests blood for certain disease from which approximately one person in a hundred suffers. People come to the clinic in group of 50. The operator of the clinic wonders whether he can increase the efficiency of the testing procedure by conducting pooled tests. In the pooled tests, the operator would pool the 50 blood samples and test them altogether. If the polled test was negative, he could pronounce the whole group healthy. If not, he could then test each person’s blood individually. The expected number of tests the operator will have to perform if he pools the blood samples are:

1 person in every 100 suffers from the disease.

Probability of a person being healthy = $$\ \frac{\ 99}{100}$$

In a group of 50 people if the test is positive, then he could then test each person’s blood individually otherwise he will consider that the entire group is healthy

The number of tests =50+1 = 51

The probability that all the people in the group are healthy = $$^{50}C_{50}\ \times\ \left(\ \frac{\ 99}{100}\right)^{50}$$

= approx 0.605

So the probability that atleast one person suffers in a group of 50 = 1-0.605= 0.395

Expected number of tests = 51*0.395 + 0.605*1

= 20.145+0.605

=20.75 = 21 tests.

C is the correct answer.