Explanation

We use the formula for the probability of the union of two events:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Where:

• $P(A \cup B) = 0.60$ (probability of qualifying for discount)
• $P(A) = 0.15$ (probability of having sprinkler system)
• $P(B) = x$ (probability of living within 5 miles of fire station - what we're solving for)
• $P(A \cap B) = 0$ (since events are mutually exclusive)

Since the events are mutually exclusive, $P(A \cap B) = 0$, so the formula simplifies to:

$P(A \cup B) = P(A) + P(B)$

$0.60 = 0.15 + x$

$x = 0.60 - 0.15$

$x = 0.45 = 45\%$

Therefore, the probability that a randomly selected homeowner lives within 5 miles of a fire station is 45%.