Question

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?
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:

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