Question

the two - way table shows the results of a recent study on the effectiveness of the flu vaccine. what is the probability that a randomly selected person who tested positive for the flu is vaccinated?

Explanation:

Step1: Recall the formula for conditional probability

Conditional probability formula is \(P(A|B)=\frac{n(A\cap B)}{n(B)}\). Here, event \(A\) is "vaccinated" and event \(B\) is "tested positive for the flu".

Step2: Identify the values from the table

\(n(A\cap B) = 465\) (number of vaccinated people who tested positive), \(n(B)=950\) (total number of people who tested positive).

Step3: Calculate the probability

\(P=\frac{465}{950}\) is incorrect. The correct formula application: \(P=\frac{465}{465 + 485}=\frac{465}{950}\) is wrong. Wait, no: For conditional probability \(P(\text{vaccinated}|\text{positive})=\frac{\text{vaccinated and positive}}{\text{total positive}}\). From the table, vaccinated and positive is \(465\), total positive is \(465+485 = 950\).

Answer:

\(\frac{465}{950}\)