Sure! Let’s tackle this step-by-step to find the probability that a student who has a dog also has a cat.
1. Understanding the Table:
The table shows the number of students who fall into different categories regarding owning cats and dogs.
[tex]\[
\begin{array}{|c|c|c|}
\hline
& \text{Has a cat} & \text{Does not have a cat} \\
\hline
\text{Has a dog} & 2 & 5 \\
\hline
\text{Does not have a dog} & 3 & 13 \\
\hline
\end{array}
\][/tex]
2. Identifying the Students with Dogs:
We need to find those students who have a dog. This includes:
- Students who have both a cat and a dog: [tex]\(2\)[/tex]
- Students who have a dog but not a cat: [tex]\(5\)[/tex]
3. Calculating the Total Number of Students with Dogs:
Add the two categories of students who have dogs:
[tex]\[
\text{Total students with a dog} = 2 + 5 = 7
\][/tex]
4. Identifying the Students with Both a Dog and a Cat:
From the table, we see that the number of students who have both a cat and a dog is [tex]\(2\)[/tex].
5. Calculating the Probability:
The probability that a student who has a dog also has a cat can be found by dividing the number of students who have both a cat and a dog by the total number of students who have a dog:
[tex]\[
\text{Probability} = \frac{\text{Number of students with both a cat and a dog}}{\text{Total number of students with a dog}} = \frac{2}{7}
\][/tex]
6. Converting the Fraction to Decimal Form:
[tex]\[
\frac{2}{7} \approx 0.2857142857142857
\][/tex]
So, the probability that a student who has a dog also has a cat is approximately [tex]\(0.2857\)[/tex] or [tex]\(28.57\%\)[/tex].
