Question

In a class of 28 students, 11 have a brother and 9 have a sister. There are 2 students

who have a brother and a sister. What is the probability that a student has a sister
given that they have a brother?

Answer 1

`\displaystyle P(A\mid B)=(2)/(11)`

Explanation:

Conditional Probability

Is a measure of the probability of the occurrence of an event, given that another event has already occurred. If event B has occurred, then the probability that event A occurs is given by:

`{\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}}`

Where
`P(A\cap B)` is the probability that both events occur and P(B) is the probability that B occurs.

Let's define events A and B for the question. There are 11 students that have a brother and 9 that have a sister. Two of those students have a brother and a sister.

We are given the fact that the selected student has a brother: this is the event that has already occurred, thus:

B = A student has a brother

A = A student has a sister

The probability that a student has a brother is:

`\displaystyle P(B)=(11)/(28)`

The probability that the student has a brother and a sister is:

`\displaystyle P(A\cap B)=(2)/(28)`

Thus, the conditional probability is:

`{\displaystyle P(A\mid B)={\frac {(2)/(28)}{(11)/(28)}}}`

Simplifying:

`\mathbf{\displaystyle P(A\mid B)=(2)/(11)}`