Q.3.7 - A First Course in Probability

Question

The king comes from a family of $2$ children. What is the probability that the other child is his sister? 

Step-by-Step Solution

Answer

The king comes from a family of $2$ children. $\frac{2}{3}$ is the probability that the other child is his sister

Step 1:Given Information

The king comes from a group of $2$children.

The group of two children comprises of the accompanying $4$ prospects:

$\left\{\right(B,B),(B,G),(G,B),(G,G\left)\right\}$

Out of these $4$, just three kinds of families comprise the occasion that the king comes from a group of two children. Those are:

$\left\{\right(B,B),(B,G),(G,B\left)\right\}$

Out of these three, just two kinds of families comprise the occasion that the king comes from a family in which the other kid is his sister. Those are:

$\left\{\right(B,G),(G,B\left)\right\}$

Step 2:Explanation

Allow $A$being the occasion that the king comes from a group of $2$ children.

Allow $B$ being the occasion that the other kid is the sister.

Obviously from the above conversation, it has,

$P\left(AB\right)=P(\text{ the king comes from a family in which the other child is his sister) }$

$=\frac{2}{4}$

$P\left(A\right)=P\left(\text{ the king comes from a family of }2\text{ children }\right)$

$=\frac{3}{4}$

Step 3:Final Answer

Presently, the conditional probability that the other youngster his sister given that the king comes from a group of two kids.

That is,

$P(B\mid A)=\frac{P\left(AB\right)}{P\left(A\right)}$

$=\frac{2/4}{3/4}$

$=\frac{2}{3}$