Boys and Girls

Probability Level 4

Consider a family of $n$ children. Define the events $A$ and $B$ as follows.
$A$ is the event that the family has both boys and girls.
$B$ is the event that he family has at most 1 girl.
Find the value of $n$ such that events $A$ and $B$ are independent.

Assumptions and Clarifications

Probability that a randomly selected child is a boy or girl is same.
Two events are independent if occurrence of one does not affect the probability of the other.

The answer is 3.

1 solution

Andreas Wendler
Apr 21, 2016

For A we get $P(A)=\sum_{k=1}^{n-1}{n \choose k}(\frac{1}{2})^{k}(\frac{1}{2})^{n-k}=(2^{n}-2)(\frac{1}{2})^{n}$ .

$P(B)=(n+1)(\frac{1}{2})^{n}$ [for 0 or 1 girl]

$P(A*B)=n(\frac{1}{2})^{n}$ [for exactly 1 girl]

Now we check independency P(A B)=P(A) P(B): $n=(n+1)(2^{n}-2)(\frac{1}{2})^{n}$

From this equation we calculate the solution $n=\boxed{3}$ .

Boys and Girls

The answer is 3.

1 solution

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