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 had only a cat, 89 had only a fish and 18 said they had all three. What is the probability that a random pet owner had exactly one pet if we know they had at least one pet?

We are asked to find the probability that a random pet owner has exactly one pet, given that they have at least one pet. Let’s break down the problem step by step.

### Step 1: Define the sets
Let:
- \( D \) be the set of pet owners who have a dog.
- \( C \) be the set of pet owners who have a cat.
- \( F \) be the set of pet owners who have a fish.

We are given the following information:
- Total pet owners surveyed: \( 984 \)
- Pet owners with a dog: \( |D| = 372 \)
- Pet owners with a cat: \( |C| = 448 \)
- Pet owners with a fish: \( |F| = 211 \)
- Pet owners with only a dog: \( 236 \)
- Pet owners with only a cat: \( 275 \)
- Pet owners with only a fish: \( 89 \)
- Pet owners with all three pets: \( 18 \)

### Step 2: Calculate the number of pet owners with exactly two pets

To find the number of people with exactly two pets, we use the inclusion-exclusion principle. First, we know:
- People with exactly one pet are given directly: \( 236 \) (only a dog), \( 275 \) (only a cat), \( 89 \) (only a fish).

To calculate the number of people with exactly two pets:
- Let \( x_{DC} \) be the number of people who have both a dog and a cat but not a fish.
- Let \( x_{DF} \) be the number of people who have both a dog and a fish but not a cat.
- Let \( x_{CF} \) be the number of people who have both a cat and a fish but not a dog.

Using the inclusion-exclusion principle, we can calculate the total number of pet owners who have at least one pet, which helps to subtract those with more than one pet later.

But for simplicity

Calculate the probability that a random pet owner had exactly one pet, based on a survey of 984 pet owners. This problem uses set theory, probability concepts, and the inclusion-exclusion principle. Perfect for high school students (Grades 9-12).

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