A study of automobile accidents produced the following data:

Model year	Proportion of all vehicles	Probability of involvement in an accident
2014	0.16	0.05
2013	0.18	0.02
2012	0.2	0.03
Other	0.46	0.04

An automobile from one of the model years 2014, 2013, and 2012 was involved in an accident. Calculate the probability that the model year of this automobile is 2014.

• 0.22
• 0.30
• 0.33
• 0.45
• 0.50

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

Solution: D

Let B, C, and D be the events of an accident occurring in 2014, 2013, and 2012, respectively. Let A = B ∪ C ∪ D .

[[math]] \operatorname{P}[B | A] = \frac{P[ A | B]P[ B]}{P[ A | B][ P[ B] + P[ A | C ]P[C ] + P[ A | D]P[ D]} [[/math]]

Use Bayes’ Theorem

[[math]] \frac{(0.05)(0.16)}{(0.05)(0.16) + (0.02)(0.18) + (0.03)(0.20)} = 0.45. [[/math]]

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