Question

testing for a disease can be made more efficient by combining samples. if the samples from three people are combined and the mixture tests negative, then all three 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 0.2, find the probability of a positive result for three 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 a negative sample

The probability of a single - sample testing positive is $p = 0.2$. So the probability of a single - sample testing negative is $q=1 - p=1 - 0.2 = 0.8$.

Step2: Find probability of all three samples being negative

Since the samples are independent, the probability that all three samples are negative is $q\times q\times q=q^{3}$. Substituting $q = 0.8$, we get $0.8^{3}=0.512$.

Step3: Find probability of a positive result

The probability of a positive result for the combined sample is the complement of the event that all three samples are negative. Let $P(X\geq1)$ be the probability of a positive result. Then $P(X\geq1)=1 - P(X = 0)$. Since $P(X = 0)=0.512$, we have $P(X\geq1)=1 - 0.512 = 0.488$.

Answer:

$0.488$