Consider a family of n children. Let two events A and B are defined as follows: A : is the event that the family has both boys and girls B : is the event that the family has atmost one girl If the events A and B are independent, then find the value of n . [Note: Probability that a randomly selected child is a boy or girl is same.]

Step-by-step solution

aligned P(A) & = ( ^n C₁+ ^n C₂+ ^n C₃+ + ^n C_ n-1 ) ( 1 2 )^n= 2^n-2 2^n P(B) & =P(0 girl or 1 girl )= ( ^n C₀+ ^n C₁ ) ( 1 2 )^n= n+1 2^n P(A B) & =P( exactly one girl )= ^n C₁ ( 1 2 )^n aligned Now, P(A B)=P(A) P(B) n 2^n = 2^n-2 2^n ( n+1 2^n ) n= (2^n-2 )(n+1) 2^n Hence, n+1 n = 2^n 2^n-2 =1+ 2 2^n-2 Hence, 1 n = 2 2^n-2 2^n-2=2 n n=3