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In a class of 29 students, 10 have a cat and 12 have a dog. There are 9 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?
Answer:

In a class of $29$ students, $10$ have a cat and $12$ have a dog. There are $9$ 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? $\newline$ Answer:

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Experimental probability

Full solution

Q. In a class of

29

students,

10

have a cat and

12

have a dog. There are

9

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?

\newline

Answer:

Determine Students with Both: Determine the number of students who have both a cat and a dog. $\newline$ We are given: $\newline$ - Total students = $29$ $\newline$ - Students with a cat = $10$ $\newline$ - Students with a dog = $12$ $\newline$ - Students without a cat or dog = $9$ $\newline$ To find the number of students who have both a cat and a dog, we can use the principle of inclusion-exclusion. The formula is: $\newline$ Number of students with both = (Students with a cat) + (Students with a dog) - (Total students) + (Students without either) $\newline$ Let's calculate this: $\newline$ Number of students with both = $10 + 12 - 29 + 9$
Calculate Both Students: Calculate the number of students who have both a cat and a dog. $\newline$ Number of students with both $= 10 + 12 - 29 + 9$ $\newline$ Number of students with both $= 22 - 29 + 9$ $\newline$ Number of students with both $= -7 + 9$ $\newline$ Number of students with both $= 2$
Calculate Probability: Calculate the probability that a student who has a dog also has a cat. $\newline$ We now know that $2$ students have both a cat and a dog. To find the probability that a student who has a dog also has a cat, we divide the number of students who have both by the total number of students who have a dog. $\newline$ The probability is given by: $\newline$ $P(\text{Has cat} | \text{Has dog}) = \frac{\text{Number of students with both}}{\text{Number of students with a dog}}$ $\newline$ Let's calculate this: $\newline$ $P(\text{Has cat} | \text{Has dog}) = \frac{2}{12}$
Simplify Probability: Simplify the probability. $\newline$ $P(\text{Has cat} | \text{Has dog}) = \frac{2}{12}$ $\newline$ $P(\text{Has cat} | \text{Has dog}) = \frac{1}{6}$

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