- Identify the frequency of the intersection between row \(C\) and column \(Y\), which is \(15\).
- Identify the total frequency for column \(Y\), which is \(30\).
- Apply the conditional probability formula: \(P(C \mid Y) = \frac{N(C \cap Y)}{N(Y)} = \frac{15}{30}\).
- The final probability is \(\boxed{0.5}\).
Explanation
- Understand the concept of conditional probability
To find the conditional probability \(P(C \mid Y)\), we need to understand that this represents the likelihood of event \(C\) occurring, given that we are restricting our focus solely to the occurrences of event \(Y\). In terms of frequency tables, this means we look only at the total count of column \(Y\) and determine what portion of that total consists of the intersection of row \(C\) and column \(Y\).
The formula for conditional probability is given by:
\[P(C \mid Y) = \frac{N(C \cap Y)}{N(Y)}\]
where \(N(C \cap Y)\) is the frequency of the intersection of row \(C\) and column \(Y\), and \(N(Y)\) is the total frequency for column \(Y\).
- Identify the values from the table
Looking at the provided table, we can identify the following values:
- The value at the intersection of row \(C\) and column \(Y\) is \(N(C \cap Y) = 15\).
- The total frequency for column \(Y\) is \(N(Y) = 30\).
These values allow us to calculate the probability by focusing only on the data in column \(Y\). We are essentially asking, "Of the 30 total cases in column \(Y\), how many fall into category \(C\)?".
- Calculate the probability
Now, we apply the values to our conditional probability formula:
\[P(C \mid Y) = \frac{15}{30}\]
Performing the division, we get:
\[P(C \mid Y) = 0.5\]
Since we are asked to round to the nearest tenth, and the result \(0.5\) is already in the tenth decimal place, we have our final value. This corresponds to the second option provided in the problem.
Examples
Conditional probability is essential for real-world decision making, such as evaluating medical test results. If a test for a disease has a certain probability of being positive, given that a patient actually has the disease, doctors use this to interpret test results correctly. For example, if a clinic performs 100 tests and 20 are positive among the 40 people who have the disease, the conditional probability of a positive test given the disease is \(P(\text{Positive} \mid \text{Disease}) = \frac{20}{40} = 0.5\).