Question

Some cars turn out to be "lemons" but, the quality of a new car is something that cannot be ascertained until after the car has been driven several hundred miles. The unlucky purchaser of a lemon may then try to unload it, by resale to someone else. Let’s assume that 10% of all new cars are lemons, and that 90% of all lemons and 5% of all non-lemons are offered for sale within the first year of ownership. If so, what fraction of all cars offered for sale within the first year are lemons?

Answer 1

Two thirds of the cars offered for sale within the first year are expected to be lemons.

Explanation:

The fraction of cars that are offered for sale within the first year and are lemons can be calculated as follow:

`F_L=(P_L)/(P_L+P_(NL))`

The proportion of lemons for sale within the first year is the product of the proportion of lemons within the new cars (10%) and the proportion that are offered for sale, given that they are lemons (90%):

`P_L=P_(Lnew)*P(Sale|L)=0.10*0.90=0.09`

The proportion of no-lemons that are for sale is the product of the proportion of no-lemons within new cars (90%) and the proportion of this group that are offered for sale (5%):

`P_(NL)=P_(NLnew)*P(Sale|NL)=0.90*0.05=0.045`

So the fraction becomes

`F_L=(P_L)/(P_L+P_(NL))=(0.09)/(0.09+0.045)= 0.667=2/3`