in a survey of 984 ﻿pet owners, 372 ﻿said they had a dog, 448 ﻿said they had a cat, and 211 ﻿said they had a fish. If 236 ﻿of the pet owners had only a dog, 275 ﻿only had a cat, 89 ﻿had only fish and 18 ﻿said they had all three. what is the probability that a pet owner had exactly 2 pets? explain like you are teaching a child

Alright, let's think about this in a fun way. Imagine you are at a big party with 984 people, and each person has pets! Some people have dogs, some have cats, some have fish, and some lucky people have more than one kind of pet.

The key here is to find out how many people at the party have **exactly two pets**. Then, we can figure out the probability that a person has exactly two pets, like finding how likely it is for someone at this party to have exactly two pets.

### Step 1: Understand the "exactly two pets" situation

At the party:
- Some people have **only one type of pet**: dog, cat, or fish.
- Some people have **all three** pets.
- And some people have **exactly two pets**, which is what we're interested in.

The people who have exactly two pets could fall into three groups:
- People with **a dog and a cat** but no fish.
- People with **a dog and a fish** but no cat.
- People with **a cat and a fish** but no dog.

### Step 2: Use the information we know

From the party (survey), we know:
- **236** people have only a dog.
- **275** people have only a cat.
- **89** people have only fish.
- **18** people have all three pets (dog, cat, and fish).

### Step 3: Calculate the people with exactly two pets

Now, we need to figure out how many people have exactly two pets in each group. Here's how:

1. **Dog and Cat (but no fish):**
   - Let's call the total number of people who have both a dog and a cat "x."
   - Out of these, **18** people have all three pets, so they are not in this group.
   - So, the number of people who have only a dog and a cat is \( x - 18 \).
   
   To find \( x \), we can use the fact that **372 people said they had a dog** and **448 people said they had a cat**. We'll use this in our calculations later.

2. **Dog and Fish (but no cat):**
   - Similar to above, we'll figure out how many people have both a dog and a fish but no cat.

3. **Cat and Fish (but no dog):**
   - Finally, we'll find out how many people have both a cat and a fish but no dog.

### Step 4:

In a survey of 984 pet owners, 372 said they had a dog, 448 said they had a cat, and 211 said they had a fish. If 236 of the pet owners had only a dog, 275 only had a cat, 89 had only fish, and 18 said they had all three, what is the probability that a pet owner had exactly 2 pets?

Explore how to calculate the probability of pet owners having exactly two pets in a survey of 984 individuals, using set theory and the inclusion-exclusion principle. This problem includes key data points such as 372 dog owners, 448 cat owners, and 211 fish owners, and guides you through the process of determining probabilities.

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