in a class of students, the following data table summarizes how many students have a brother or a sister…

Explanation:

Step1: Calculate the number of students who have a brother

The number of students who have a brother is the sum of those who have a brother and a sister ($9$) and those who have a brother but no sister ($2$). So, $9 + 2=11$.

Step2: Calculate the number of students who have a brother and a sister

The number of students who have a brother and a sister is $9$.

Step3: Use the formula for conditional probability

The formula for conditional probability $P(A|B)=\frac{P(A\cap B)}{P(B)}$. In the case of frequency - based probability (where $n(X)$ is the number of elements in set $X$), if $A$ is the event of having a sister and $B$ is the event of having a brother, then $P(A|B)=\frac{n(A\cap B)}{n(B)}$. Substituting $n(A\cap B) = 9$ and $n(B)=11$, we get the probability.

Answer:

The probability that a student has a sister given that they have a brother is $\frac{9}{11}$.