To find the probability that a student has a dog given that they have a cat, we follow these steps:
1. Identify the relevant data from the table:
- The number of students who have both a cat and a dog: [tex]\(6\)[/tex]
- The number of students who have a cat but not a dog: [tex]\(9\)[/tex]
2. Calculate the total number of students who have a cat:
- This is the sum of students who have a cat and a dog, and students who have a cat but not a dog:
[tex]\[
\text{Total students who have a cat} = 6 + 9 = 15
\][/tex]
3. Apply the formula for conditional probability:
- The probability that a student has a dog given that they have a cat is given by the ratio of the number of students who have both a cat and a dog to the total number of students who have a cat.
[tex]\[
P(\text{Dog} | \text{Cat}) = \frac{\text{Number of students who have both a cat and a dog}}{\text{Total number of students who have a cat}}
\][/tex]
4. Substitute the values into the formula:
- Number of students who have both a cat and a dog: [tex]\(6\)[/tex]
- Total number of students who have a cat: [tex]\(15\)[/tex]
[tex]\[
P(\text{Dog} | \text{Cat}) = \frac{6}{15} = 0.4
\][/tex]
Therefore, the probability that a student has a dog given that they have a cat is [tex]\(0.4\)[/tex] or [tex]\(40\%\)[/tex].
