Question

In a class of students, the following data table summarizes how many students have a cat or a dog. What is the probability that a student has a cat given that they do not have a dog? ( NO LINKS ) ​

Answer 1

Given:

Number of students who has a cat and a dog = 5

Number of students who has a cat but do not have a dog = 11

Number of students who has a dog but do not have a cat = 3

Number of students who neither have a cat nor a dog = 2

To find:

The probability that a student has a cat given that they do not have a dog.

Solution:

Let the following events:

A = Student has a cat

B = Do not have a dog

Total number of outcomes is:

$5+3+11+2=21$

The probability that a student has a cat but do not have a dog is:

$P(A\cap B)=\dfrac{11}{21}$

The probability that a student do not have a dog is:

$P(B)=\dfrac{11+2}{21}$

$P(B)=\dfrac{13}{21}$

The conditional probability is:

$P\left(\dfrac{A}{B}\right)=\dfrac{P(A\cap B)}{P(B)}$

$P\left(\dfrac{A}{B}\right)=\dfrac{\dfrac{11}{21}}{\dfrac{13}{21}}$

$P\left(\dfrac{A}{B}\right)=\dfrac{11}{13}$

Therefore, the probability that a student has a cat given that they do not have a dog is $\dfrac{11}{13}$.