Question

In a class of 26 students, 17 have a brother and 12 have a sister. There are 5 students who do not have any siblings. What is the probability that a student has a brother given that they have a sister?

Answer 1

The number of students have a brother is 17.

The number of students have a sister is 12.

The number of students who donot have any siblings are 5.

The total students are 26.

Let there are x students who have both brother and sister. So number of students with only brother s (17 - x), number of students with sister only is (12 - x).

Determine the number of students have both brother and sister.

$\begin{gathered} 17-x+12-x+x+5=26 \\ -x=26-17-12-5 \\ x=8 \end{gathered}$

So probability for students with both brother and sister is,

$P(\text{ Brother and sister)=}\frac{8}{26}$

Probability for student have sister is,

$P(\text{ sister)=}\frac{12}{26}$

Determine the probability that a student has a brother given that they have a sister.

$\begin{gathered} P\text{ (Brother and sister|sister)=}\frac{P\text{ (Brother and sister)}}{P\text{ (Sister)}} \\ =\frac{\frac{8}{26}}{\frac{12}{26}} \\ =\frac{8}{26}\cdot\frac{26}{12} \\ =\frac{8}{12} \\ =\frac{2}{3} \end{gathered}$

Answer: 2/3