The number of traffic accidents occurring on any given day in Coralville is Poisson distributed with mean 5. The probability that any such accident involves an uninsured driver is 0.25, independent of all other such accidents.

Calculate the probability that on a given day in Coralville there are no traffic accidents that involve an uninsured driver.

• 0.007
• 0.010
• 0.124
• 0.237
• 0.287

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

Solution: E

From the Law of Total Probability, the required probability is

[[math]] \begin{align*} \sum_{k=0}^{\infty} P(\textrm{0 accidents with an insured driver} | \textrm{k accidents})P( \textrm{k accidents}) &= \sum_{k=0}^{\infty} (0.75)^k \frac{e^{-5}5^k}{k!} \\ &= \frac{e^{-5}}{e^{-3.75}} \sum_{k=0}^{\infty} \frac{e^{-3.75}(3.75)^k}{k!} \\ &= e^{-1.25} \\ &= 0.287. \end{align*} [[/math]]

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