Suppose that I have the set {01, b, c, d, e, f, g, h, 1'}. Without repeating digits, a) How many 6-Ietter \"words\" (i.e. unique rearrangements of letters) can I make from this set? b) Of those 6letter words, how many contain exactly one vowel? A class of 20 students wants to elect a committee consisting of a president, a vice president, a treasurer, and 4 secretaries. How many different committees can be formed? (Note: The secretary positions are all identical.) How many bit strings (i.e. sequences of Us and ls) of length 15 begin with 010 OR end with 101 (or both)? Suppose I am dealt a 5-card hand, from a regular deck of 52 cards. What is the probability that I am dealt: a) at least three cards with the same value e.g. (9,9,9,3,4), (4,4,4,4,5), etc. b) at least two cards with the same value e.g. (2,2,4,5,6), (Q,Q,Q,2,3), (2,2,4,4,5), etc. 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