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 have a dog?

Explanation:

Step1: Recall conditional probability formula

The formula for conditional probability is \( P(A|B)=\frac{P(A\cap B)}{P(B)} \). In the context of a two - way table, if \( A \) is "has a cat" and \( B \) is "has a dog", then the number of students who have both a cat and a dog is the number of students in the cell where "Has a dog" and "Has a cat" intersect, and the number of students who have a dog is the sum of the numbers in the "Has a dog" row.

Step2: Identify the relevant numbers

From the table, the number of students who have a dog and a cat (\( n(A\cap B) \)) is 3. The number of students who have a dog (\( n(B) \)) is the sum of the number of students who have a dog and a cat and the number of students who have a dog and not a cat. So \( n(B)=3 + 17=20 \).

Step3: Calculate the probability

Using the formula for probability (since we are dealing with counts, the probability \( P(A|B)=\frac{n(A\cap B)}{n(B)} \)), we substitute the values we found. So \( P(\text{has a cat}|\text{has a dog})=\frac{3}{3 + 17}=\frac{3}{20} \).

Answer:

\(\frac{3}{20}\)