in a class of 28 students, 10 have a brother and 12 have a sister. there are 10 students who do not have any…

Explanation:

Step1: Find number of students with siblings

Total students are 28 and 10 have no siblings. So number of students with siblings is $28 - 10=18$.

Step2: Find number of students with only a sister

Let $A$ be students with a brother and $B$ be students with a sister. We know $n(A) = 10$, $n(B)=12$ and $n(A\cup B)=18$. Using the formula $n(A\cup B)=n(A)+n(B)-n(A\cap B)$, we get $18 = 10+12 - n(A\cap B)$, so $n(A\cap B)=4$. Then number of students with only a sister is $n(B)-n(A\cap B)=12 - 4 = 8$.

Step3: Find number of students without a brother

Number of students without a brother is $28 - 10=18$.

Step4: Calculate the conditional - probability

The formula for conditional probability is $P(B|\overline{A})=\frac{n(\overline{A}\cap B)}{n(\overline{A})}$. Here, $n(\overline{A}\cap B)$ is the number of students without a brother and with a sister (i.e., only a sister), which is 8, and $n(\overline{A}) = 18$. So $P(B|\overline{A})=\frac{8}{18}=\frac{4}{9}$.

Answer:

$\frac{4}{9}$