Question

In a class of students, the following data table summarizes how many students have abrother or a sister. What is the probability that a student has a brother given that theyhave a sister?Has a brɔther Does not have a brotherHas a sister26Does not have a sister78

Answer 1

The conditional probability formula is:

`\begin{gathered} P(A|B)=(P(A\cap B))/(P(B)) \\ \text{ Where} \\ P(A|B)\text{ is the probability of A given B,} \\ P(A\cap B)\text{ is the probability of A and B, and} \\ P(B)\text{ is the probability of B} \end{gathered}`

On the other hand, the probability formula is:

`P(A)=\frac{\text{ Number of favourable outcomes to A}}{\text{ Total number of possible outcomes}}`

To solve the exercise, we first complete the table:

So, in this case, we have:

`\begin{gathered} P(\text{ Student has a brother and they have a sister})=(2)/(23) \\ P(\text{They have a sister})=(8)/(23) \\ P(\text Student has a brother )=((2)/(23))/((8)/(23)) \\ P(\text They have a sister)=(2\cdot23)/(8\cdot23) \\ P(\text They have a sister)=(2)/(8) \\ \text{ Simplify} \\ P(\text Student has a brother )=(1\cdot2)/(4\cdot2) \\ P(\text Student has a brother )=(1)/(4) \end{gathered}`

Therefore, the probability that a student has a brother given that they have a sister is 1/4.