Question

In a class of students, the following data table summarizes how many students have a cat or a dog. What is the probability that a student has a cat given that they do not have a dog?
Has a cat Does not have a cat
Has a dog 5 4
Does not have a dog 8 7
Answer
Attempt 1 out of 2

Answer 1

To find the probability that a student has a cat given that they do not have a dog, we can use the formula for conditional probability:

\[
P(A | B) = \frac{P(A \cap B)}{P(B)}
\]

Where:
- \(A\) is the event that a student has a cat.
- \(B\) is the event that a student does not have a dog.

From the data table:

- Students who **have a cat and do not have a dog**: 8
- Students who **do not have a dog**: This includes those who have a cat (8) plus those who do not have a cat (7). So, \(8 + 7 = 15\).

Now we can plug these values into the conditional probability formula:

\[
P(\text{Has a cat} | \text{Does not have a dog}) = \frac{\text{Number of students with a cat and no dog}}{\text{Total number of students with no dog}} = \frac{8}{15}
\]

Thus, the probability that a student has a cat given that they do not have a dog is:

\[
\frac{8}{15}
\]

So the final answer is \(\frac{8}{15}\).