2. Motor vehicles sold to individuals are classified as either cars or light tru

Question

2. Motor vehicles sold to individuals are classified as either cars or light trucks (including SUVs) and year, 69%ofvelilssold ) and as either domestic or imported. In a recent year, 69% of vehicles sold were light trucks, 78% were domestic, and 55% were domestic light trucks. (a) What is the probability that a sold vehicle is a light truck or domestic? (b) Given that a sold vehicle is imported, what is the probability that it is also a light truck? (c) Are the events "vehicle is a light truck" and "vehicle is imported" independent. Justify your answer.

Explanation / Answer

Here we are given that 69% of the vehicles sold were light trucks. Therefore P( light trucks ) = 0.69. This means that P( cars) = 1 - P( light trucks ) = 1 - 0.69 = 0.31

Also we are given that 78% of the motor vehicles were domestic. Therefore P( domestic ) = 0.78

Therefore P( imported) = 1 - P( domestic ) = 1 - 0.78 = 0.22

Also we are given that 55% are domestic light trucks. Therefore P( domestic and light ) = 0.55

Now the 2 x 2 table could formed here as:

a) Probability that a sold vehicle is a light truck or domestic is computed from the table as:

= 0.78 + 0.14 = 0.92

Therefore 0.92 is the required probability here.

b) Given that vehicle is imported, probability that it is also a light truck is computed using the Bayes theorem as:

= P( imported and light truck ) / P( imported )

= 0.14 / 0.22

= 0.6364

Therefore 0.6364 is the required probability here.

c) P( light truck) = 0.69 and P( imported ) = 0.22

Therefore, P( light truck) P( imported ) = 0.69*0.22 = 0.1518

Also P( light truck and imported ) = 0.14

Therefore, P( light truck) P( imported ) is not equal to P( light truck and imported )

Therefore the 2 events are not independent.

Cars Light Trucks Domestic 0.23 0.55 0.78 Imported 0.08 0.14 0.22 0.31 0.69 1

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