A GRE - Probability Question

Question

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?

Answer 1

$$A = \underbrace{P(G \cup H)}_{\text{either $G$ or $H$}} - \underbrace{P(G\cap H)}_{\text{but not both}} = P(G) + P(H) - P(G\cap H) - P(G\cap H) = r + s - 2rs$$ 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. $\Box$

Answer 2

As another way of looking at it:

Probability that G occurs but not H is $r(1-s)$

and Probability that H occurs but not G is $s(1-r)$.

Adding gives $r+s-2rs$ which is less than $r+s-rs$.