Financial Risk Manager Part 1

Explanation

We are given:

• Let A = event that a homeowner has a sprinkler system
• Let B = event that a homeowner lives within 5 miles of a fire station
• P(A ∪ B) = 0.60 (60% qualify for the discount)
• P(A) = 0.15 (15% have a sprinkler system)
• P(A ∩ B) = 0 (none of those with sprinkler systems live within 5 miles of a fire station)

Using the probability addition rule:

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

Substituting the known values:

$`$0.60` = 0.15 + P(B) - 0$$

$P(B) = 0.60 - 0.15 = 0.45$

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

Key points:

• The problem uses the addition rule for probability
• The events are mutually exclusive for those with sprinkler systems (P(A ∩ B) = 0)
• The solution involves simple algebraic rearrangement