The probability that a member of a certain class of homeowners with liability and property coverage will file a liability claim is 0.04, and the probability that a member of this class will file a property claim is 0.10. The probability that a member of this class will file a liability claim but not a property claim is 0.01.

Calculate the probability that a randomly selected member of this class of homeowners will not file a claim of either type.

• 0.850
• 0.860
• 0.864
• 0.870
• 0.890

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Step 1: Define Events and Given Probabilities

Let [math]L[/math] be the event that a member files a liability claim. Let [math]P[/math] be the event that a member files a property claim. The problem provides the following probabilities:

Given Probabilities

Event Description	Probability
Probability of filing a liability claim	[math]P(L) = 0.04[/math]
Probability of filing a property claim	[math]P(P) = 0.10[/math]
Probability of filing a liability claim but not a property claim	[math]P(L \cap P^c) = 0.01[/math]

Step 2: Calculate the Probability of Filing Both Claims

The event of filing a liability claim ([math]L[/math]) can be split into two mutually exclusive events: filing a liability claim and a property claim ([math]L \cap P[/math]), or filing a liability claim but not a property claim ([math]L \cap P^c[/math]). Therefore, we can write:

[[math]]P(L) = P(L \cap P) + P(L \cap P^c)[[/math]]

We are given [math]P(L) = 0.04[/math] and [math]P(L \cap P^c) = 0.01[/math]. We can solve for [math]P(L \cap P)[/math]:

[[math]]0.04 = P(L \cap P) + 0.01[[/math]]

[[math]]P(L \cap P) = 0.04 - 0.01 = 0.03[[/math]]

Thus, the probability that a member files both a liability and a property claim is 0.03.

Step 3: Calculate the Probability of Filing Only a Property Claim

Similarly, the event of filing a property claim ([math]P[/math]) can be split into two mutually exclusive events: filing a property claim and a liability claim ([math]L \cap P[/math]), or filing a property claim but not a liability claim ([math]P \cap L^c[/math]). Therefore, we can write:

[[math]]P(P) = P(L \cap P) + P(P \cap L^c)[[/math]]

We are given [math]P(P) = 0.10[/math] and we calculated [math]P(L \cap P) = 0.03[/math] in the previous step. We can solve for [math]P(P \cap L^c)[/math]:

[[math]]0.10 = 0.03 + P(P \cap L^c)[[/math]]

[[math]]P(P \cap L^c) = 0.10 - 0.03 = 0.07[[/math]]

So, the probability that a member files only a property claim (and not a liability claim) is 0.07.

Step 4: Calculate the Probability of Not Filing Either Claim

We need to find the probability that a randomly selected member will not file a claim of either type. This is represented by [math]P(L^c \cap P^c)[/math]. Using De Morgan's laws, [math]P(L^c \cap P^c) = P((L \cup P)^c)[/math]. This means the probability of not filing either claim is 1 minus the probability of filing at least one claim. The event of filing at least one claim ([math]L \cup P[/math]) can be decomposed into the sum of the probabilities of three mutually exclusive events we have already determined:

Components of P(L ∪ P)

Event	Probability
Filing a liability claim but not a property claim	[math]P(L \cap P^c) = 0.01[/math]
Filing both a liability and a property claim	[math]P(L \cap P) = 0.03[/math]
Filing a property claim but not a liability claim	[math]P(P \cap L^c) = 0.07[/math]

The probability of filing at least one claim is the sum of these probabilities:

[[math]]P(L \cup P) = P(L \cap P^c) + P(L \cap P) + P(P \cap L^c)[[/math]]

[[math]]P(L \cup P) = 0.01 + 0.03 + 0.07 = 0.11[[/math]]

Finally, the probability of not filing either claim is:

[[math]]P(\text{Neither}) = 1 - P(L \cup P)[[/math]]

[[math]]P(\text{Neither}) = 1 - 0.11 = 0.89[[/math]]

Key Insights

• Understanding probability notation for events and their complements ([math]P(A)[/math], [math]P(A^c)[/math]) and intersections ([math]P(A \cap B)[/math]).
• Decomposition of a probability into mutually exclusive parts: [math]P(A) = P(A \cap B) + P(A \cap B^c)[/math]. This allows solving for unknown intersections.
• The complement rule for probabilities: [math]P(E^c) = 1 - P(E)[/math]. This is essential for finding the probability of "not filing either claim".
• De Morgan's Laws for sets: [math]P(A^c \cap B^c) = P((A \cup B)^c)[/math], which is useful for translating "neither A nor B" into "not (A or B)".
• The probability of the union of mutually exclusive events is the sum of their individual probabilities.

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Solution: E

Liability but not property = 0.01 (given)

Liability and property = 0.04 – 0.01 = 0.03.

Property but not liability = 0.10 – 0.03 = 0.07

Probability of neither = 1 – 0.01 – 0.03 – 0.07 = 0.89

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.