Question

the probability of a randomly selected adult in one country being infected with a certain virus is 0.004. in tests for the virus, blood samples from 29 people are combined. what is the probability that the combined sample tests positive for the virus? is it unlikely for such a combined sample to test positive? note that the combined sample tests positive if at least one person has the virus. the probability that the combined sample will test positive is (round to three decimal places as needed.)

Explanation:

Step1: Find the probability of a person not having the virus

The probability of a person being infected \(p = 0.004\). So the probability of a person not being infected \(q=1 - p=1 - 0.004 = 0.996\)

Step2: Find the probability that all 29 people are not infected

Since the events (each person not being infected) are independent, for \(n = 29\) people, the probability that all are not infected is \(q^{n}\). Using the formula \(P(\text{all negative})=(0.996)^{29}\)

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Step3: Find the probability that the combined sample tests positive

The probability that the combined sample tests positive (at least one positive) is \(P(\text{at least one positive})=1 - P(\text{all negative})\)

$$P(\text{at least one positive})=1-(0.996)^{29}\approx1 - 0.890=0.110$$

Answer:

\(0.110\)