Question

when doing blood testing for a viral infection, the procedure can be made more efficient and less expensive by combining partial samples of different blood specimens. if samples from four people are combined and the mixture tests negative, we know that all four individual samples are negative. find the probability of a positive result for four samples combined into one mixture, assuming the probability of an individual blood sample testing positive for the virus is 0.02. round the answer to four decimal place accuracy.

Explanation:

Step1: Find probability of negative result for one sample

The probability of an individual sample testing positive is $p = 0.02$. So the probability of an individual sample testing negative is $q=1 - p=1 - 0.02 = 0.98$.

Step2: Use the multiplication rule for independent events

Since the samples are independent, the probability that all four samples are negative (and thus the mixture is negative) is $q^4$. Substitute $q = 0.98$ into the formula: $(0.98)^4=0.98\times0.98\times0.98\times0.98 = 0.9224$.

Step3: Find probability of positive result for the mixture

The probability of a positive result for the mixture is the complement of the probability of a negative result for the mixture. Let $P(X = 1)$ be the probability of a positive - result for the mixture. Then $P(X = 1)=1-(0.98)^4=1 - 0.9224=0.0776$.

Answer:

$0.0776$