Question

You would like to determine who in a group of 100 students carries antibodies for a certain virus. You can perform blood tests on each student individually, which would require 100 tests. Instead, you can partition the students into 10 groups of 10. Combine the blood samples of the 10 students in each group, and analyze the combined sample. If none of the 10 students in that group carries the antibodies, the test will show negative, while if one or more do carry the antibodies, the test will turn out positive, and you could then test every student in that group individually, resulting in a total of 11 tests for that group. If each person has the antibodies with probability .1, independently of each other,

find: (a) The maximum number of tests you may need to perform.

(b) The expected number of tests you’ll perform.

(c) Explain in words whether you would raise or lower the group sizes when the antibody probability is close to 0 or 1.

Answer 1

(a) The maximum number of tests that we need to perform is when the antibody tests turn out positive for all the 10 groups = Number of tests for 1 group*Total groups = 11*10 = 110

(b) Now, Each group has 10 people such that each person has antibodies with probability 0.1, INDEPENDENT OF EACH OTHER.

Therefore, Probability that atleast one of the members of the group has antibodies (p) = 1 - (Probability that no member has a antibody) = 1 - (0.9)^10 = 0.651321559.

{Antibody probability for each person (P) = 0.1}

{Probability that 10 people don't have antibody = (1 - 0.1)^10 = 0.9^10 (As the are independent)}

{n = 10 for 10 groups}

The Expected Number of groups that turn out positive for the antibody test = n*p = 10*(1-(0.9^10)) = 6.513215599.

The Expected number of tests that you need to perform = 11*(Expected number of groups that turn out positive for the antibody test) = 11*6.5132155997 = 71.64537159.

(c) If the antibody probability for each person (P) is close to 0, then the value of p will also be close to 0. Therefore, for a small number of groups the probability that the groups turn out positive will be less as compared to that of large groups. Therefore, to increase the expected value of groups that turn out positive, we will need to increase the number of groups so that the probability of finding a particular group as positive will increase. So, the number of people in the groups will decrease to increase the probability of finding a particular group as positive.

Therefore, we will lower the group sizes when the antibody probability is close to zero.

If P is close to 1, then the value of p will also be close to 1. Therefore, for a small number of groups there will always be a significant expected number of groups that turn out positive for the antibody test. Therefore, we can conduct the test even if there are a large number of people in a group because the probability of finding a particular group as positive will still be significant if more number of people are present in that group.

Therefore, we will raise the group sizes when the antibody probability is close to one.

Do comment for any doubts.