In a class of students, the following data table summarizes how many students have a brother or a sister. What is the probability that a student has a sister given that they do not have a brother?

Question

Has a brother Does not have a brother
Has a sister 8 12
Does not have a sister 5 4
Answer

Answer 1

To find the probability that a student has a sister given that they do not have a brother, we will use the formula for conditional probability:

\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]

Where:

• \(A\) is the event that a student has a sister.
• \(B\) is the event that a student does not have a brother.

From the data provided in the table, we can summarize the counts:

• Number of students who have a sister and do not have a brother: \(12\)
• Number of students who do not have a brother: This includes:
  • Has a sister and does not have a brother: \(12\)
  • Does not have a sister and does not have a brother: \(4\)

So the total who do not have a brother is:

\[ 12 + 4 = 16 \]

Now, we need to determine \(P(A \cap B)\), which is the number of students who have a sister and do not have a brother:

\[ P(A \cap B) = 12 \]

Now we can calculate \(P(A | B)\):

\[ P(A | B) = \frac{P(A \cap B)}{P(B)} = \frac{12}{16} = \frac{3}{4} \]

Thus, the probability that a student has a sister given that they do not have a brother is:

\[ \frac{3}{4} \text{ or } 0.75 \]