Probability of Having a Pet in Households

In a city, there are 1200 households and a survey revealed that 800 households have at least one pet. Among these, 500 households have dogs, 400 have cats, and 300 have both dogs and cats. If a household is randomly selected from the city, what is the probability that it has a pet?

Let’s denote:

• Let $D$ be the set of households with dogs.
• Let $C$ be the set of households with cats.

We know:

• Number of households with dogs, $|D| = 500$
• Number of households with cats, $|C| = 400$
• Number of households with both dogs and cats, $|D \cap C| = 300$

Using the principle of inclusion-exclusion, the number of households with at least one type of pet can be calculated as:

$$|D \cup C| = |D| + |C| - |D \cap C|$$

Then, determine the probability of choosing a household with at least one pet from the total number of households, which is:

$$P(pet) = \frac{|D \cup C|}{\text{Total Households}}$$