In a class of 30 students, 8 have a brother and 13 have a sister. There are 3 students who have a brother and a sister. What is the probability that student who does not have a brother has a sister?
Let's denote the events as follows:
A: student has a brother
B: student has a sister
We have P(A) = 8/30 = 4/15 (students who have a brother)
P(B) = 13/30 (students who have a sister)
From the given information, there are 3 students who have both a brother and a sister. Therefore, this event is P(A and B) = 3/30 = 1/10.
Now, we want to find the probability that a student who does not have a brother has a sister, which is represented by P(B| A'). A' represents the complement of having a brother.
P(B|A') = P(B and A') / P(A') = (P(B) - P(A and B)) / (1 - P(A)) = [13/30 - 1/10] / (1 - 4/15) = (27/30) / (11/15) = 27/30 * 15/11 = 9/22
Therefore, the probability that a student who does not have a brother has a sister is 9/22.