A large number N of people are subjected to a blood investigation. Thi



Answered question


A large number N of people are subjected to a blood investigation. This investigation can be organized in two ways.
(1) The blood of each person is investigated separately. In this case N analyses are needed.
(2) The blood of k people are mixed and the mixture is analysed. If the result is negative, then this single analysis is sufficient for k persons. But if it is positive, then the blood of each one must be subsequently investigated separately, and in toto for k people, k+1 analysis are needed. It is assumed that the probability of a positive result (p) is the same for all people and that the results of the analysis are independent in the probabilistic sense.
For what k is the minimum expected number of necessary analysis attained?

Answer & Explanation

Tamara Donohue

Tamara Donohue

Beginner2021-11-18Added 11 answers

Let N represents negative, P represents positive
In a sample of k people, the test comes negative only if none of the k people's blood tests positive.
So, P(N)=(1p)k
The total test population is divided into N/k groups of k people each
Expected Number of test, E(X) is:
Now, for the minimum number of tests, differentiate above equation w.r.t. k,
We will equate this result to zero for minimum condition,
Approximating the logarithms by ln(1p)p for small p, we obtain
Solving this quadratic, we get
For small p, both solutions are close and given by

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Multivariable calculus

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?