Financial Risk Manager Part 1
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A homeowners insurer offers a discount for homeowners that either have a sprinkler system or live within 5 miles of a fire station. 60% of homeowners qualify for the discount. Only 15% of homeowners have a sprinkler system, and none of those homeowners live within 5 miles of a fire station. If the events are mutually exclusive, what is the probability a randomly selected homeowner lives within 5 miles of a fire station?
Explanation:
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%.