Question

the test to detect the presence of respiratory syncytial virus is 97% accurate for a person who has the virus and 97% accurate for a person who does not have the virus. in a given population, 0.55% of the people are infected. what is the probability that a person does have the respiratory syncytial virus given they have a negative result on their test? a. 98.46% b. 99.45% c. 0.02% d. 1.65%

Explanation:

Step1: Define events

Let \(V\) be the event that a person has the virus, and \(\overline{V}\) be the event that a person does not have the virus. Let \(N\) be the event that the test result is negative.

We know \(P(V)=0.0055\), \(P(\overline{V}) = 1 - P(V)=0.9945\), \(P(N|V)=1 - 0.97 = 0.03\) (probability of false - negative), \(P(N|\overline{V})=0.97\) (probability of true - negative)

Step2: Apply Bayes' theorem

Bayes' theorem is \(P(V|N)=\frac{P(N|V)P(V)}{P(N|V)P(V)+P(N|\overline{V})P(\overline{V})}\)

Substitute the values:

$$
LATEXBLOCK0
$$

Answer:

C. \(0.02\%\)