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Stat 330 (Spring 2015): Homework 1 Due: January 23, 2015 Show all of your work, and please staple your assignment if you use more than one sheet. Write your name, the course number and the section on every sheet. Problems marked with * will be graded and one additional randomly chosen problem will be graded. 1. A coin is tossed three times, and the sequence of heads and tails is recorded. (a) Determine the sample space, Ω. (b) List the elements that make up the following events: i. A = at least two heads, ii. B = the first two tosses result in heads, iii. C = the last toss results in a tail (c) List the elements of the following events: i. A, ii. A ∩ C. iii. A ∪ C 2. * Let A, B ⊂ Ω. Draw Venn diagrams that shows the sets A ∪ B, A ∩ B, and A ∩ B̄. (Many drawings are correct, depending on your choices of A and B.) Then, use Kolmogorov’s axioms to show that the following statements are true. (a) For any two events A, B ⊂ Ω, P (A ∩ B̄) = P (A) − P (A ∩ B) (b) For two events A, B ⊂ Ω with B ⊂ A, P (A ∩ B̄) = P (A) − P (B) (c) For events A, B, C ⊂ Ω verify the following extension of the addition rule: P (A ∪ B ∪ C) = P (A) + P (B) + P (C) − P (A ∩ B) − P (A ∩ C) − P (B ∩ C) + P (A ∩ B ∩ C) 3. How many 6-character license plate combinations are possible if there must be three letters followed by three numbers? 4. In how many ways can 12 people be divided into three groups of 4 for an evening of bridge? In how many ways can this be done if the 12 consist of six pairs of partners that must play with each other? 5. A drawer of socks contains seven black socks, eight blue socks, and nine green socks. Two socks are chosen in the dark. (a) What is the probability that they match? (b) What is the probability that a black pair is chosen? 6. * (Baron’s book): 2.1 7. (Baron’s book): 2.2 8. (Baron’s book): 2.4 1