Probability
Probability
Allen Star Batch
Grade 12
Question:
A family has three children. Event 'A' is that the 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. Then:
Events 'A' and 'B' are independent
Events 'A' and 'B' are not independent
Events $A, B, C$ are not independent
Events $A, B, C$ are independent
Step-by-Step Solution
Key Concept: Three events are independent if all pairwise and triple intersections satisfy the multiplication rule; failure of any condition means dependence.
$P(A) = P(C)$ where event $A$ is no boy or exactly one boy: $P(A) = \left(\frac{1}{2}\right)^3 + ^3C_1\left(\frac{1}{2}\right)^3 = \frac{1}{8} + \frac{3}{8} = \frac{1}{2}$. Event $B$ is 2 boys, 1 girl or 1 boy, 2 girls: $P(B) = ^3C_1\left(\frac{1}{2}\right)^3 + ^3C_2\left(\frac{1}{2}\right)^3 = \frac{3}{4}$. Event $C$ is no girl or exactly one girl: $P(C) = \frac{1}{2}$. Since $P(A \cap B) = \frac{3}{8} = P(A) \times P(B)$ fails and $A \cap C = \emptyset$, events $A$, $B$, $C$ are not mutually independent.
Correct Answer: 1,4