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In a class of $30$ students, $18$ have a cat and $21$ have a dog. There are $5$ students who do not have a cat or a dog. What is the probability that a student who has a dog also has a cat?

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Probability of independent and dependent events

Full solution

Q. In a class of

30

students,

18

have a cat and

21

have a dog. There are

5

students who do not have a cat or a dog. What is the probability that a student who has a dog also has a cat?

Find Students with Both Pets: First, let's find out how many students have both a cat and a dog. We know the total number of students is $30$ , and $5$ students don't have either pet. So, the number of students with at least one pet is $30 - 5 = 25$ .
Use Inclusion-Exclusion Principle: Using the principle of inclusion-exclusion to find the number of students who have both pets. We add the number of students who have a cat and those who have a dog, then subtract the total number of students who have at least one pet: $18$ (students with a cat) + $21$ (students with a dog) - $25$ (students with at least one pet) = $14$ . So, $14$ students have both a cat and a dog.

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