An actuary studied the likelihood that different types of drivers would be involved in at least one collision during any one-year period. The results of the study are:

Type of Driver	Percentage of all drivers	Probability of at least one collision
Teen	8%	0.15
Young adult	16%	0.08
Midlife	45%	0.04
Senior	31%	0.05

Given that a driver has been involved in at least one collision in the past year, calculate the probability that the driver is a young adult driver.

• 0.06
• 0.16
• 0.19
• 0.22
• 0.25

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

Solution: D

Let

[math]C[/math] = Event of a collision

[math]T[/math] = Event of a teen driver

[math]Y[/math] = Event of a young adult driver

[math]M[/math] = Event of a midlife driver

[math]S[/math] = Event of a senior driver

Then using Bayes’ Theorem, we see that

[[math]] \begin{align*} \operatorname{P}[Y|C] &= \frac{\operatorname{P}[C | Y ]\operatorname{P}[Y ]}{\operatorname{P}[C | T ]\operatorname{P}[T ] + \operatorname{P}[C | Y ]\operatorname{P}[Y ] + \operatorname{P}[C | M ]\operatorname{P}[ M ] + \operatorname{P}[C | S ]\operatorname{P}[ S ]} \\ &= \frac{(0.08)(0.16)}{(0.15)(0.08) + (0.08)(0.16) + (0.04)(0.45) + (0.05)(0.31)} \\ &= 0.22 \end{align*} [[/math]]

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