A and B are two independent events. The probability that both events A and B occur is \(\displaystyle\frac{1}{6}\) and the probability that neither of them occur is \(\displaystyle\frac{1}{3}\). If \(P(A) = x, P(B) = y\) then the value of \(x+y\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{5}{6}\)
“Neither A nor B” is the complement of \(A \cup B\), so
\( \displaystyle P(A \cup B) = 1 - \frac{1}{3} = \frac{2}{3} \)
By the addition theorem, \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\):
\( \displaystyle \frac{2}{3} = x + y - \frac{1}{6} \)
\( \displaystyle x + y = \frac{2}{3} + \frac{1}{6} = \frac{5}{6} \)
(Independence is not needed for x + y; it gives \(xy = \frac{1}{6}\), so \(P(A)\) and \(P(B)\) are \(\frac{1}{2}\) and \(\frac{1}{3}\).)