Question

Testing for a disease can be made more efficient by combining samples. If the samples from two people are combined and the mixture tests​ negative, then both samples are negative. On the other​ hand, one positive sample will always test​ positive, no matter how many negative samples it is mixed with. Assuming the probability of a single sample testing positive is 0.15​, find the probability of a positive result for two samples combined into one mixture. Is the probability low enough so that further testing of the individual samples is rarely​ necessary?

Answer 1

Solution;

Let A be the positive event in a single sample;

$P(A)=0.15$

But;

$P(X=x)=(\displaystyle_x^n)p^x(1-p)^{n-x}$

The probability of positive results in two combined samples is;

$P(X>0)=1-P(X=0)$

Since;

$P(X=0)=(\displaystyle_0^2)0.15^0(1-0.15)^2$

$P(X=0)=0.7225$

Therefore;

$P(X>0)=1-0.7225=0.2775$

The probability is low therefore no necessary further testing