The annual numbers of thefts a homeowners insurance policyholder experiences are analyzed over three years.

Define the following events:

1. A = the event that the policyholder experiences no thefts in the three years.
2. B = the event that the policyholder experiences at least one theft in the second year.
3. C = the event that the policyholder experiences exactly one theft in the first year.
4. D = the event that the policyholder experiences no thefts in the third year.
5. E = the event that the policyholder experiences no thefts in the second year, and at least one theft in the third year.

Determine which three events satisfy the condition that the probability of their union equals the sum of their probabilities.

• Events A, B, and E
• Events A, C, and E
• Events A, D, and E
• Events B, C, and D
• Events B, C, and E

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

Solution: A

The probability a union of three events equals the sum of their probabilities if and only if they are mutually exclusive, that is, no two of them can both occur. Events A and B cannot both occur since no thefts in the first three years would imply no thefts in the second year, thus precluding the possibility of at least 1 theft in the second year. Events A and E cannot both occur since no thefts in the first three years would imply no thefts in the third year, thus precluding the possibility of at least 1 theft in the third year. Events B and E cannot both occur since it is impossible to experience both no thefts and at least 1 theft in the second year. Thus, events A, B, and E satisfy the desired condition.

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