Question

testing for a disease can be made more efficient by combining samples. if the samples from four people are combined and the mixture tests negative, then all four 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 0.1, find the probability of a positive result for four samples combined into one mixture. is the probability low enough so that further testing of the individual samples is rarely necessary?
the probability of a positive test result is
(round to three decimal places as needed.)

Explanation:

Step1: Find probability of all - negative samples

The probability that a single sample is negative is $1 - 0.1=0.9$. Since the samples are independent, the probability that all 4 samples are negative is $(0.9)^4$.
$(0.9)^4 = 0.9\times0.9\times0.9\times0.9= 0.6561$.

Step2: Find probability of a positive - result mixture

The probability of a positive - result mixture is the complement of the probability that all samples are negative. Let $P(X\geq1)$ be the probability of a positive - result mixture and $P(X = 0)$ be the probability that all samples are negative. Then $P(X\geq1)=1 - P(X = 0)$.
$P(X\geq1)=1-(0.9)^4=1 - 0.6561 = 0.344$.

Answer:

$0.344$