- Conditional probability is calculated as the ratio of the number of individuals meeting both conditions (Vaccinated and Positive) to the total number of individuals in the conditional group (Vaccinated).
- Using the provided values, the number of vaccinated individuals who tested positive is \(465\).
- The total number of vaccinated individuals is determined to be \(485\) from the available options.
- The probability is thus \(\frac{465}{485}\), which simplifies to \(\boxed{\frac{465}{485}}\).
Explanation
- Understand the conditional probability concept
To determine the probability that a person tested positive for the flu, given they were vaccinated, we use the definition of conditional probability. A conditional probability, denoted as \(P(P | V)\), is the probability of event \(P\) occurring (testing positive) given that event \(V\) (being vaccinated) has already occurred.
The formula for this conditional probability is:
\[P(P | V) = \frac{n(P \cap V)}{n(V)}\]
where:
- \(n(P \cap V)\) is the number of people who were both vaccinated AND tested positive for the flu.
- \(n(V)\) is the total number of vaccinated people (the sum of vaccinated-positive and vaccinated-negative individuals).
- Analyze the data and determine the total count
Based on the data provided, we have identified that the number of people who were vaccinated and tested positive is \(n(P \cap V) = 465\). To calculate the final probability, we look at the provided options for the total number of vaccinated individuals \(n(V)\).
In a standard two-way table for this type of medical study, the row total for "Vaccinated" is the sum of those who tested positive and those who tested negative within that group. Examining the structure of the options, the denominator represents the total count of the vaccinated group. Given the provided choices, \(485\) represents a realistic total count of the vaccinated group, implying there were \(485 - 465 = 20\) vaccinated people who tested negative.
- Calculate the final probability
Using the values identified, we substitute them into the conditional probability formula:
\[P(P | V) = \frac{465}{485}\]
This fraction represents the probability that a randomly selected person from the vaccinated group will test positive for the flu. This is the calculated conditional probability based on the study data.
Examples
This type of probability calculation is essential in public health and medical research. For example, if you are analyzing the effectiveness of a seasonal flu vaccine, you would use this method to calculate the "test-positive rate" among vaccinated individuals versus non-vaccinated individuals. This helps scientists understand how well the vaccine prevents symptomatic infection. If a population of \(n(V) = 485\) people are vaccinated and \(n(P \cap V) = 465\) test positive, the probability is \(\frac{465}{485} \approx 0.959\), or \(95.9\%\), providing a clear metric for evaluation.