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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