## Number of Children in a Family Independent Events Problem

In a family of
$n$
children let A be the event that'at most one child is a boy' and let event B be the event 'every child is the same sex'. For what value(s) of
$n$
are A and B independent?
Assuming that having a boy or girl are equally likely and equal to 1/2, then the number of boys in the family,
$X$
, can be modelled by a binomial distribution,
$X \sim B(n, 1/2)$
..
$P(A)=P(X=0)+P(X=1)={}^nC_0 (\frac{1}{2})^0 (\frac{1}{2})^n+{}^nC_n (\frac{1}{2})^1(\frac{1}{2})^{n-1}=(1+n) (\frac{1}{2})^n$
.
If at most child in the family is a boy and every child in the family is the same sex, then there can either be only one child, either a boy or girl( with probability 1), or every child must be a girl.
$P(X=1)=\frac{1}{2}$
and
$n=1$

$X=0=(\frac{1}{2})^n$
for any
$n$

Suppose
$n=1$
then
$(1+1) (\frac{1}{2})^1=1$
so
$n=1$
is possible.
Suppose
$n \gt 1$
then
$(n+1)(\frac{1}{2})^n = (\frac{1}{2})^n$
which has no solutions for
$n \gt 1$
so
$n=1$
is the only value that makes A and B independent.

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