Question

in a study, 1,085 out of 2,321 people did not receive a flu vaccination. 465 people were vaccinated and tested positive for the flu. a total of 1,371 participants tested negative. create a two - way table for this scenario. use the two - way table to determine the probability of randomly selecting a person who tested positive for the flu, given he or she was vaccinated. \\(\frac{465}{2,321}\\) \\(\frac{465}{1,236}\\) \\(\frac{465}{950}\\) \\(\frac{465}{485}\\) positive negative total vaccinated 465 771 1,236 not vaccinated 485 600 1,085 total 950 1,371 2,321 correct! you have completed this task.

Explanation:

Step1: Recall Conditional Probability Formula

The formula for conditional probability is \( P(A|B)=\frac{P(A\cap B)}{P(B)} \). In this case, event \( A \) is "tested positive" and event \( B \) is "was vaccinated". So we need the number of people who were vaccinated and tested positive (which is \( n(A\cap B) \)) divided by the number of people who were vaccinated (which is \( n(B) \)).

Step2: Identify the Numbers from the Table

From the two - way table, the number of people who were vaccinated and tested positive (\( n(A\cap B) \)) is 465. The number of people who were vaccinated (\( n(B) \)) is the total number of vaccinated people, which is 1236 (from the "Total" column of the "Vaccinated" row).

Step3: Calculate the Probability

Using the formula for conditional probability (in terms of counts, since we are dealing with a sample space of people in the study), the probability \( P(\text{Positive}|\text{Vaccinated})=\frac{\text{Number of vaccinated and positive}}{\text{Number of vaccinated}}=\frac{465}{1236} \)

Answer:

\(\frac{465}{1236}\) (corresponding to the option \(\boldsymbol{\frac{465}{1236}}\))