- The problem asks for the conditional probability of being vaccinated, given a positive flu test result.
- We restrict the sample space to the total number of positive test results, which is \(950\).
- We identify the count of people who are both vaccinated and tested positive as \(465\).
- The resulting probability is \(\boxed{\frac{465}{950}}\).
Explanation
- Analyze the provided data from the two-way table
The objective is to find the probability that a person is vaccinated, given that they tested positive for the flu. This is a classic conditional probability problem. The information provided in the two-way table is as follows:
- Vaccinated and positive: \(465\)
- Vaccinated and negative: \(771\)
- Vaccinated total: \(1,236\)
- Not vaccinated and positive: \(485\)
- Not vaccinated and negative: \(600\)
- Not vaccinated total: \(1,085\)
- Total positive: \(950\)
- Total negative: \(1,371\)
- Grand total: \(2,321\)
- Identify the sample space for the condition
When we are looking for a conditional probability, such as "the probability that someone is vaccinated, GIVEN that they tested positive," our sample space is restricted to only those who met the condition. In this case, the condition is that the person tested positive for the flu. From the table, the total number of people who tested positive is \(950\). This will be the denominator of our probability fraction.
- Identify the favorable outcomes
Within our restricted sample space of people who tested positive (which is \(950\)), we need to find how many of those individuals are vaccinated. Looking at the "Vaccinated" row and the "Pos." column in the table, we see that there are \(465\) people who are both vaccinated and tested positive. This value will be the numerator of our probability fraction.
- Apply the conditional probability formula and conclude
The conditional probability formula is defined as:
\[P(\text{Vaccinated} | \text{Positive}) = \frac{P(\text{Vaccinated and Positive})}{P(\text{Positive})} = \frac{\text{Number of Vaccinated and Positive}}{\text{Total Number of Positive}}\]
By substituting our identified values into the formula, we get:
\[P(\text{Vaccinated} | \text{Positive}) = \frac{465}{950}\]
Comparing this result to the given options, we can see that it matches the third option provided.
Examples
Conditional probability is used in real-life scenarios like medical diagnosis or insurance risk assessment. For example, if you are testing for a specific condition, the probability of having that condition given a positive test result helps doctors understand the reliability of the test. In this case, the calculation \(P(\text{Vaccinated} | \text{Positive}) = \frac{465}{950}\) helps us quantify the proportion of vaccinated individuals among those who tested positive.