Probability that group 1 was chosen for selection, given what the selected item turned out to be I have this question in exercise which I solved it but not sure about the answer. ### Question There are two department in a car manufacturing company, each department is making cars. Department 1 has 10 racing cars and 30 luxury cars, while department 2 has 20 of each. A marketing manager picks a department at random, then picks a car at random. The car turns out to be a luxury car. How probable is it that the manager picked it out of department 1? ### Answer Probability of picking a luxury car from department 1 $= 1/2*(30/40)=0.375$ Probability of picking a luxury car from department 2 $= 1/2*(20/40)=0.25$ More probable than picking up from department 2.

Use Bayes' rule:


P(Dep1|Rac) = P(Rac|Dep1)P(Dep1)/(P(Dep1)P(Rac|Dep1) + P(Dep2)P(Rac|Dep2))


where

P(Dep1|Rac): probability the state of the world is Dep1 given a racing car was selected--unknown

P(Rac|Dep1): probability a racing car is selected given the department is Dep1--0.25

P(Dep1): prior probability Dep1 is chosen--0.5

P(Rac|Dep1): probability of choosing a racing car given it is from Dep1--0.25

P(Dep2): prior probability Dep2 is chosen--0.5

P(Rac|Dep2): probability of choosing a racing car given it is from Dept2--0.5

Substituting and doing the arithmetic gives

P(Dep1|Rac) = 1/3.