A driver and a passenger are in a car accident. Each of them independently has probability 0.3 of being hospitalized. When a hospitalization occurs, the loss is uniformly distributed on [0, 1]. When two hospitalizations occur, the losses are independent. Calculate the expected number of people in the car who are hospitalized, given that the total loss due to hospitalizations from the accident is less than 1.

• 0.510
• 0.534
• 0.600
• 0.628
• 0.800

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

Solution: B

The unconditional probabilities for the number of people in the car who are hospitalized are 0.49, 0.42 and 0.09 for 0, 1 and 2, respectively. If the number of people hospitalized is 0 or 1, then the total loss will be less than 1. However, if two people are hospitalized, the probability that the total loss will be less than 1 is 0.5. Thus, the expected number of people in the car who are hospitalized, given that the total loss due to hospitalizations from the accident is less than 1 is

[[math]] \frac{0.49}{0.49 + 0.42 + 0.09 ⋅ 0.5} (0) + \frac{0.42}{0.49 + 0.42 + 0.09 ⋅ 0.5} (1) + \frac{0.09 ⋅ 0.5}{0.49 + 0.42 + 0.09 ⋅ 0.5} (2) = 0.534. [[/math]]

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