Question 10

Scenario: A certain system can experience three different types of defects.

Events:

  • where - A defect of type .

Probabilities:

  • .
  • .
  • .
  • .
  • .
  • .
  • .

(a)

Problem: Given that the system has a type 1 defect, what is the probability that it has a type 2 defect?

Process: The question can be rephrased to ask what the probability is of the system having a type defect given that it has a type defect. This would be the conditional probability . To find the conditional probability, we need to find the probability of the intersection (and) of and .

Answer: 0.6.

(b)

Problem: Given that the system has a type 1 defect, what is the probability that it has all three types of defects?

Process: The question can be rephrased to ask what the probability is of the system having a type and defect given that it has a type defect. This would be the conditional probability . We are already given the probability of the intersection of , , and .

Answer: 0.1.

(c)

Problem: Given that the system has at least one type of defect, what is the probability that it has exactly one type of defect?

Process: The question can be rephrased to ask what the probability is of the system having exactly one type of defect given that it has at least one type of defect. The phrase "at least one type of defect" indicates we're finding the probability of the union (or) of all three types of defects. The conditional probability that we're going to find is the sum of , , and . To find all three conditional probabilities, we need to find the probability of the union of all three types of defects. In order to find that probability, we need to find the probability of the intersection (and) of each possible pair you can make between the types of defects.

Answer: 0.3077.

(d)

Problem: Given that the system has both of the first two types of defects, what is the probability that it does not have the third type of defect?

Process: The question can be rephrased to ask what the probability is of the system not having a type 3 defect given that it has a type 1 and type 2 defect. This would be the conditional probability . In this case, isn't just the probability of the complement of . .

Answer: 0.8333.