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In a class of 30 students, 10 have a cat and 21 have a dog. There are 2 students who do not have a cat or a dog. What is the probability that a student has a cat given that they do not have a dog?
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In a class of $30$ students, $10$ have a cat and $21$ have a dog. There are $2$ students who do not have a cat or 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

30

students,

10

have a cat and

21

have a dog. There are

2

students who do not have a cat or 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 do not have a dog. Since there are $2$ students who do not have a cat or a dog, we subtract this number from the total number of students to find the number of students who have a dog or a cat but not a dog. $\newline$ Total number of students = $30$ $\newline$ Number of students with a dog = $21$ $\newline$ Number of students with neither a cat nor a dog = $2$ $\newline$ Number of students without a dog = Total number of students - Number of students with a dog $\newline$ Number of students without a dog = $30 - 21$ $\newline$ Number of students without a dog = $9$
Find Students with Only a Cat: Next, we need to find out how many of those students without a dog have a cat. Since there are $10$ students with a cat and $2$ students with neither a cat nor a dog, we subtract the number of students with neither from the total number of students with a cat to get the number of students with only a cat. $\newline$ Number of students with a cat $= 10$ $\newline$ Number of students with neither a cat nor a dog $= 2$ $\newline$ Number of students with only a cat $=$ Number of students with a cat $-$ Number of students with neither a cat nor a dog $\newline$ Number of students with only a cat $= 10 - 2$ $\newline$ Number of students with only a cat $= 8$
Calculate Probability: 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. $\newline$ Probability = $\frac{\text{Number of students with only a cat}}{\text{Number of students without a dog}}$ $\newline$ Probability = $\frac{8}{9}$

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