Number of Children in a Family Independent Events Problem
\[n\]
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. 
