Financial Risk Manager Part 1

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%.