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Solve the following.
The probability of the event A occurring is 1/3 and of the event B occurring is 1/2. If A and B are…
The probability of the event $A$ occurring is $\frac{1}{3}$ and of the event $B$ occurring is $\frac{1}{2}$. If $A$ and $B$ are independent events, then find the probability of neither $A$ nor $B$ occurring.
Answer
Answer
$P(A) = \frac{1}{3}$, $P(B) = \frac{1}{2}$, and $A$, $B$ are independent, so $A'$ and $B'$ are independent.
$P(A') = \frac{2}{3}$, $P(B') = \frac{1}{2}$
$P(\text{neither } A \text{ nor } B) = P(A')P(B') = \frac{2}{3} \times \frac{1}{2} = \frac{1}{3}$
Hence the probability is $\frac{1}{3}$.
Final answer: $\frac{1}{3}$