The waiting time for the first claim from a good driver and the waiting time for the first claim from a bad driver are independent and follow exponential distributions with means 6 years and 3 years, respectively.

Calculate the probability that the first claim from a good driver will be filed within 3 years and the first claim from a bad driver will be filed within 2 years.

• [math]\frac{1}{18}(1 − e^{−2/3} − e^{−1/2} + e^{−7/6} )[/math]
• [math]\frac{1}{18} e^{−7/6} [/math]
• [math]1 − e^{−2/3} − e^{−1/2} + e^{−7/6}[/math]
• [math]1 − e^{−2/3} − e^{−1/2} + e^{−1/3} [/math]
• [math]1 − \frac{1}{3}e^{−2/3} − \frac{1}{6}e^{−1/2} + \frac{1}{18}e^{−7/6} [/math]

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

Solution: C

For a good driver, the probability is [math]1 − e^{−3/6}[/math] and for a bad driver, the probability is [math]1 − e^{−2/3}[/math] . The probability of both is the product,

[[math]](1 − e^{−3/6} )(1 − e^{−2/3 }) = 1 − e^{−1/2} − e^{−2/3} + e^{−7/6}.[[/math]]

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