Let A and B be two independent events such that \(P(A) = a\) and \(P(B) = b\). If the probability that exactly one of A and B occurs is \(\displaystyle\frac{2}{3}\), then the value of \(4a + 4b - 8ab\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{8}{3}\)
“Exactly one occurs” means A occurs and B does not, or B occurs and A does not. These two cases cannot happen together, and for independent events the probabilities multiply:
\( P(A \cap B') = a(1 - b) \)
\( P(A' \cap B) = (1 - a)b \)
Adding the two cases:
\( P(\text{exactly one}) = a - ab + b - ab \)
\( = a + b - 2ab \)
This is given to be \(\frac{2}{3}\):
\( \displaystyle a + b - 2ab = \frac{2}{3} \)
The required expression is 4 times this:
\( 4a + 4b - 8ab = 4(a + b - 2ab) \)
\( \displaystyle = 4 \times \frac{2}{3} = \frac{8}{3} \)