5. Let A,B be events, where P(A n BC) = 0.2 and P(AC n B\") = 0.3. a) What is P(B)? (Note: There is not enough information here to draw a complete Venn diagram, but you have enough information to calculate P(B)) b) Suppose now that A and B are independent. Calculate P(A n B). 6. A factory manufactures a certain part for a computer. After creation, the parts are rated for their quality: A, B or C. It is known that: 70% of all parts manufactured are rated 'A'. 20% of all parts manufactured are rated '3'. 10% of all parts manufactured are rated 'C'. It is also known that: 4% ofthe A-rated parts will fail in Year 1. 15% of the Brated parts will fail in Year 1. 25% of C-rated parts will fail in Year 1. Suppose that a given component failed in Year 1. What is the probability that it was a B- rated part? 7. Let A and B be events with non-zero probabilities. For each of the following, identify whether the statement is true or false. If it is true: Prove it. If it is false: Give a counterexample to illustrate this. a) If P(A) = P(B), then A and B are independent. b) If P(A) 2 P(B), then P(A|B) 2 P(B|A). c) If P(A|B) = 1, then P(B|A) = 1 d) If AC = B, then A and B cannot be independent