An auto company investigated car accidents for their models

An auto company investigated car accidents for their models during 2005-2013. Among all the cars under investigation, there are 10% from the 2005 models, and there are 15% from the 2006 models. The data shows that if a car came from the 2005 models (resp. 2006 models), the probability of an accident is 5% (resp. 4%). Now a car is selected at random from all 2005-2013 models. If a car from 2005 or 2006 has an accident, what’s the probability that it’s a 2005 model?

Solution

P[car facing accident is a 2005 model | car from 2005 or 2006 faced accident]

=P[car from 2005 model faced accident]/P[car from 2005 or 2006 faced accident]

=P[car from 2005 model]*P[it faced accident]/(P[car from 2005 model]*P[it faced accident]+P[car from 2006 model]*P[it faced accident])=0.1*0.05/(0.1*0.05+0.15*0.04)=0.45454