In a class of 19 students, 12 have a brother and 11 have a sister. There are 5 students who do not have any siblings. What is the probability that a student has a sister given that they do not have a brother?
Let's define the events:
Event A: A student has a brother
Event B: A student has a sister
Given that there are 12 students who have a brother and 11 students who have a sister, we can find the total number of students who have neither a brother nor a sister:
Total = Total Number of Students - Students with a Brother - Students with a Sister = 19 - 12 - 11 = 19 - 23 = 5
Now, we need to find the probability that a student has a sister given that they do not have a brother. This can be calculated using the conditional probability formula:
P(B | A') = P(B and A') / P(A')
Where P(B and A') represents the probability of a student having a sister and not having a brother, and P(A') represents the probability of not having a brother.
From the information given, we know that there are 5 students who do not have any siblings. Therefore, P(A') = 5 / 19.
Now, to find the probability of a student having a sister and not having a brother, we need to find the number of students who have a sister but not a brother. Since there are only 5 students with no siblings, this number would be 5 - students with a brother = 5 - 12 = 0.
Therefore, P(B and A') = 0 / 19 = 0
Substitute these values into the conditional probability formula:
P(B | A') = 0 / (5 / 19) = 0
Therefore, the probability that a student has a sister given that they do not have a brother is 0.