Question

A consumer organization estimates that 29% of new cars have a cosmetic defect. such as a scratch or a dent, when they are delivered to car dealers this same organization believes that 7% have a functional defect-something that does not work properly-and that 2% of new cars have both kinds of problems.

a) if you buy a new car, what's the probability that it has some kind of defect?
b) what's the probability it has a cosmetic defect but no functional defect?
c) if you notice a dent on a new car, what's the probabil- ity it has a functional defect?
d) are the two kinds of defects disjoint events? explain.
e) do you think the two kinds of defects are independent events? explain.

Answer 1

A new car has a 34% chance of having some kind of defect, a 27% chance of solely a cosmetic defect, and if a cosmetic defect is observed, there's a 6.9% chance it also has a functional defect. The two types of defects are not disjoint; there may be dependencies between them.

Step-by-step explanation:

To answer the student's questions regarding the probability of defects in new cars, we can use basic probability principles.

• a) The probability that a new car has any kind of defect is found by summing the probabilities of each kind of defect and then subtracting the probability of both occurring, to avoid double-counting. Thus, P(cosmetic or functional defect) = P(cosmetic defect) + P(functional defect) - P(both defects) = 0.29 + 0.07 - 0.02 = 0.34 or 34%.
• b) The probability that a new car has a cosmetic defect but no functional defect is the probability of a cosmetic defect minus the probability of both defects, P(cosmetic defect but not functional) = P(cosmetic defect) - P(both defects) = 0.29 - 0.02 = 0.27 or 27%.
• c) If you notice a dent on a new car, the probability that it also has a functional defect can be found using conditional probability. P(functional defect | cosmetic defect) = P(both defects) / P(cosmetic defect) = 0.02 / 0.29 ≈ 0.069 or 6.9%.
• d) The two kinds of defects are not disjoint events because cars can have both defects simultaneously, as indicated by the 2% probability of a car having both defects.
• e) The two kinds of defects being independent would mean that the occurrence of one does not affect the probability of the other. Given that 2% of cars have both defects while 29% have cosmetic and 7% have functional defects separately, it suggests a possible dependency, as the occurrence of one defect might influence the occurrence of the other.