To solve this problem, we'll break it into two parts: finding the total number of students who do not have a dog, and then finding the probability that a student has a cat given that they do not have a dog.
1. Total number of students who do not have a dog:
We look at the table rows under the column "Does not have a dog":
- Number of students who have a cat and do not have a dog: 4
- Number of students who do not have a cat and do not have a dog: 7
Adding these values together gives us the total number of students who do not have a dog:
[tex]\[
\text{Total number of students who do not have a dog} = 4 + 7 = 11
\][/tex]
2. Probability that a student has a cat given they do not have a dog:
We use the conditional probability formula [tex]\( P(\text{Cat} | \text{No Dog}) \)[/tex]:
[tex]\[
P(\text{Cat} | \text{No Dog}) = \frac{\text{Number of students who have a cat and do not have a dog}}{\text{Total number of students who do not have a dog}}
\][/tex]
Plugging in the values:
[tex]\[
P(\text{Cat} | \text{No Dog}) = \frac{4}{11}
\][/tex]
Simplifying the fraction results in:
[tex]\[
P(\text{Cat} | \text{No Dog}) \approx 0.3636 \text{ or } 36.36\%
\][/tex]
Thus, the total number of students who do not have a dog is 11, and the probability that a student has a cat given that they do not have a dog is approximately 0.3636, or 36.36%.
