To determine the probability that a student has a sister given that they have a brother, we need to use the concept of conditional probability. Conditional probability is calculated as follows:
[tex]\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \][/tex]
Where:
- [tex]\( P(A | B) \)[/tex] is the probability that event [tex]\( A \)[/tex] occurs given that event [tex]\( B \)[/tex] has occurred.
- [tex]\( P(A \cap B) \)[/tex] is the probability that both events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] occur.
- [tex]\( P(B) \)[/tex] is the probability that event [tex]\( B \)[/tex] occurs.
In this context:
- Event [tex]\( A \)[/tex] is "the student has a sister."
- Event [tex]\( B \)[/tex] is "the student has a brother."
From the given data, we can extract the following values:
- Number of students who have both a brother and a sister ( [tex]\( P(A \cap B) \)[/tex] ) = 6
- Number of students who have a brother, with or without a sister ( [tex]\( P(B) \)[/tex] ) = 6 (both brother and sister) + 2 (brother but no sister) = 8
Now, we can calculate the conditional probability that a student has a sister given that they have a brother:
[tex]\[ P(\text{sister} | \text{brother}) = \frac{\text{Number of students who have both a brother and a sister}}{\text{Total number of students who have a brother}} = \frac{6}{8} = 0.75 \][/tex]
So, the probability that a student has a sister given that they have a brother is [tex]\( 0.75 \)[/tex] or 75%.
