Answer:
To find the probability that a student chosen at random has both a cat and a dog, we need to analyze the given information and construct a Venn diagram.
Let's start by representing the class of 25 students in a Venn diagram. We will have two overlapping circles, one for cats and one for dogs. The intersection of these circles will represent the students who have both a cat and a dog.
According to the given information, 15 students have a cat and 16 students have a dog. However, we are not provided with the exact number of students who have both pets. To find this value, we need to consider the fact that 3 students have neither a cat nor a dog.
Let's denote the number of students who have both a cat and a dog as "x." Therefore, the number of students who only have a cat would be (15 - x), and the number of students who only have a dog would be (16 - x).
Now, let's construct the Venn diagram based on this information:
```
Cat
-----
| |
| |
| |
Dog ---|-----|---
| |
| |
| |
-----
```
In this diagram, the number inside each section represents the number of students falling into that category.
Since there are 3 students who have neither a cat nor a dog, we can add this information to our Venn diagram:
```
Cat
-----
| |
| |
| |
Dog ---|-----|---
| |
| 3 |
| |
-----
```
Now, let's fill in the remaining sections of the Venn diagram using the information we derived earlier:
```
Cat
-----
| |
x |(15-x)|
| |
Dog ---|-----|---
| |
| 3 |
| |
-----
```
To find the value of "x," we can use the fact that the total number of students is 25. Therefore, the sum of all sections in the Venn diagram should equal 25.
(15 - x) + x + (16 - x) + 3 = 25
Simplifying this equation, we get:
34 - x = 25
x = 34 - 25
x = 9
Now, let's update our Venn diagram with the correct values:
```
Cat
-----
| |
9 | 6 |
| |
Dog ---|-----|---
| |
| 3 |
| |
-----
```
From the Venn diagram, we can see that there are 6 students who have only a cat, and there are 9 students who have both a cat and a dog.
To find the probability that a student chosen at random has both a cat and a dog, we divide the number of students who have both pets by the total number of students:
Probability = Number of students with both cat and dog / Total number of students
Probability = 9 / 25
Therefore, the probability that a student chosen at random has both a cat and a dog is 9/25.
Explanation: