Opgavee

In a car factory, 1.5% of the vehicles have a defect in the injection system that threat- ens to destroy the engine after only a short drive. As a result, cars are tested when they leave the assembly line. However, the test in not entirely foolproof. The factory owner
knows that if a car has a fault in the injection system, then there is a 90% chance that it will fail the test. However, he also knows that there is a 1% chance that a flawless car will fail the test.
Let X be a random variable that takes the value 1 if a car has the defect and 0 if not. Let Y be another random variable that takes the value 1 if the vehicle fails the test and and the value 0 if it does not.
a) Derive the conditional probability function for Y given X, and then the marginal probability function of X, fX(x).
b) Are X and Y independent?
c) What is the probability that a vehicle that fails the test actually has a faulty inection
system? (Use Bayes’ theorem.) What is the probability of it not having the problem?

Kan nogle hjælpe med a og b