Items are inspected for flaws by two quality inspectors. Both inspectors inspect every item and the probability that an item has a flaw is 0.1. If a flaw is present, it will be detected by the first inspector with probability 0.92, and by the second inspector with probability 0.7. If an item does not have a flaw, it will be passed by the first inspector with probability 0.95 and by the second inspector with probability 0.8. Assume the inspectors function independently.

(a) If an item has a flaw, what is the probability that it will be found by at least one of the two inspectors?

(b) If an item is passed by the first inspector, what is the probability that it actually has a flaw?

(c) What is the probability that the two inspectors draw different conclusions on the same item?

(d) If an item is passed by both inspectors, what is the probability that it actually has a flaw?

In a class of students, 60%, 50% and 30% choose English, French and German, respectively. Out of the students who choose English, 40% of them also take French. None of the students who choose both English and French take German. The event of choosing German is independent of the event of taking English. The number of students who choose both English and German is twice that of students who choose both French and German. Let X be the number of subjects chosen by a student.

(a) Draw a Venn diagram showing all the information given in the question.

(b) Find the probability distribution for X.

(c) What are the values of the mean and standard deviation of X?

(d) If a student who does not choose French is selected at random, what is theprobability that he/she takes both English and German?