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In a class of 18 students, 8 have a cat and 7 have a dog. There are 3 students who have a cat and a dog. What is the probability that a student has a cat given that they do not have a dog?
Answer:

In a class of $18$ students, $8$ have a cat and $7$ have a dog. There are $3$ students who have a cat and a dog. What is the probability that a student has a cat given that they do not have a dog? $\newline$ Answer:

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Probability of independent and dependent events

Full solution

Q. In a class of

18

students,

8

have a cat and

7

have a dog. There are

3

students who have a cat and a dog. What is the probability that a student has a cat given that they do not have a dog?

\newline

Answer:

Determine Number of Students: First, we need to determine the number of students who have only a cat and not a dog. We know that $8$ students have a cat and $3$ of those also have a dog. So, the number of students who have only a cat is the total number of students with a cat minus the number of students who have both a cat and a dog. $\newline$ Calculation: $8$ (students with a cat) - $3$ (students with both a cat and a dog) = $5$ (students with only a cat).
Find Total Students Without Dog: Next, we need to find out the total number of students who do not have a dog. Since $7$ students have a dog and $3$ of those also have a cat, we subtract the number of students with a dog from the total number of students to get the number of students without a dog. $\newline$ Calculation: $18$ (total students) $-$ $7$ (students with a dog) $=$ $11$ (students without a dog).
Calculate Probability Given No Dog: Now, we can calculate the probability that a student has a cat given that they do not have a dog. This is the number of students with only a cat divided by the number of students without a dog. $\newline$ Calculation: Probability = $\frac{5 \text{ (students with only a cat)}}{11 \text{ (students without a dog)}}$ .
Calculate Probability: The probability is therefore $\frac{5}{11}.$

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