A policyholder purchases automobile insurance for two years. Define the following events:

F = the policyholder has exactly one accident in year one.

G = the policyholder has one or more accidents in year two. Define the following events:

1. The policyholder has exactly one accident in year one and has more than one accident in year two.
2. The policyholder has at least two accidents during the two-year period.
3. The policyholder has exactly one accident in year one and has at least one accident in year two.
4. The policyholder has exactly one accident in year one and has a total of two or more accidents in the two-year period.
5. The policyholder has exactly one accident in year one and has more accidents in year two than in year one.

Determine the number of events from the above list of five that are the same as [math]F \cap G [/math].

• None
• Exactly one
• Exactly two
• Exactly three
• All

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

Solution: C

1. is false because G includes having one accident in year two.
2. is false because there could be no accidents in year one.
3. is true because it connects the descriptions of F and G (noting that “one or more” and “at least one” are identical events) with “and.”
4. is true because given one accident in year one (F), having a total of two or more in two years is the same as one or more in year two (G).
5. is false because it requires year two to have at least two accidents.

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