An insurance company insures a good driver and a bad driver on the same policy. The table below gives the probability of a given number of claims occurring for each of these drivers in the next ten years.

Number of Claims	Probability for the good driver	Probability for the bad driver
0	0.5	0.2
1	0.3	0.3
2	0.2	0.4
3	0.0	0.1

The number of claims occurring for the two drivers are independent.

Calculate the mode of the distribution of the total number of claims occurring on this policy in the next ten years.

• 0
• 1
• 2
• 3
• 4

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

Solution: C

The probability needs to be calculated for each total number of claims. 0: 0.5(0.2) = 0.10 1: 0.5(0.3) + 0.3(0.2) = 0.21 2: 0.5(0.4) + 0.3(0.3) + 0.2(0.2) = 0.33 3: 0.5(0.1) + 0.3(0.4) + 0.2(0.3) + 0.0(0.2) = 0.23 At this point there is only 0.13 probability remaining, so the mode must be at 2.

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