$A$ and $B$ are two independent events. The probability that both $A$ and $B$ occurs is $1/6$ and the probability that neither of them occurs is $1/3$. Then the probability of the occurrence of $A$ may be:
Step-by-Step Solution
Key Concept: For independent events, use both the product rule $P(A ∩ B) = P(A)P(B)$ and complement properties simultaneously to set up a solvable system.
Given that $A$ and $B$ are independent events with $P(A ∩ B) = 1/6$ and $P(\bar{A} ∩ \bar{B}) = 1/3$, we use $P(A)P(B) = 1/6$ and the complement relation to establish $P(A) + P(B) = 5/6$. Setting $P(A) = x$ and $P(B) = y$, we solve the system $xy = 1/6$ and $x + y = 5/6$ to get $x = 1/2, y = 1/3$ or $x = 1/3, y = 1/2$. Therefore $P(A) = 1/2$ or $1/3$.
Correct Answer: 1,2