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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?

Respuesta :

Answer:

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

Step-by-step explanation:

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

[tex]F_L=\frac{P_L}{P_L+P_{NL}}[/tex]

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%):

[tex]P_L=P_{Lnew}*P(Sale|L)=0.10*0.90=0.09[/tex]

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%):

[tex]P_{NL}=P_{NLnew}*P(Sale|NL)=0.90*0.05=0.045[/tex]

So the fraction becomes

[tex]F_L=\frac{P_L}{P_L+P_{NL}}=\frac{0.09}{0.09+0.045}= 0.667=2/3[/tex]

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