In a probability experiment, G and H are independent events. The probability that G will occur is r, and the probability that H will occur is s, where both r and s are greater than 0.
Quantity A
The probability that either G will occur or H will occur, but not both.
Quantity B
r+s-rs
A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
The answer is B. How?
Best Answer
$$A = \underbrace{P(G \cup H)}_{\text{either $G$ or $H$}} - \underbrace{P(G\cap H)}_{\text{but not both}} = P(G) + P(H) - P(G\cap H) - P(G\cap H) = r + s - 2rs$$ By inclusion-exclusion principle. $$B = r + s - rs = P(G) + P(H) - P(G\cap H) = P(G\cup H)$$ Since $$A = r + s - 2rs < r+s-rs = B$$ We have that B) is the correct answer. $\Box$