A family has three children. Event ‘A’ is that family has at most one boy, Event ‘B’ is that family has at least one boy and one girl, Event ‘C’ is that the family has at most one girl. Find whether events ‘A’ and ‘B’ are independent. Also find whether A, B, C are independent or not.
Text Solution
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A and B are independent but A,B,C are not independent
Sol. P = P clearly undoward ; A → no boy or exactly one boy in family
⇒ P = 
+ 3 C 1 
= 
+ 
= 
B → 2 boy , 1 girl or 1 boy , 2 girl ; P = 3 C 2 
+ 3 C 1 
= 
= 
C : no girl or exactly one girl ; P(C) = 
+ 3 C 1 
= 
A ∩ B → one boy & two girls ; P (A ∩ B) = 3 C 1 
= 
B ∩ C → one girl and two boys P(B ∩ C) = 
A ∩ C → φ ⇒ A ∩ B ∩ C in also φ.
P(A ∩ B) = 
= P(A) × P(B) , so A and B are independent
neither P(A ∩ B ∩ C) = P(A) × P(B) × P(C) nor P(A ∩ C) = P(A) × P(C). So ABC are not independent
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