1. Suppose we roll a fair k-sided die with the numbers 1 through k on the dies faces. If X is the number that appears, what is E[X]? 2. Let G = (V, E) be a connected graph that is not Eulerian. Prove that it is possible to add a single vertex to G, together with some edges from this new vertex to some old vertices such that the new graph is Eulerian.

3. A committee of three people has two members each of whom independently has prob- ability p of making the correct decision and a third member who flips a fair coin for each

decision. The final verdict is decided by majority rule. There is also a one-person commit- tee that has probability p of making the correct decision. Which committee has the better

probability of making the correct decision? 4. Consider an experiment involving two successive rolls of a 4-sided die in which all 16 possible outcomes are equally likely and have probability 1/16. (a) Let Ai be the event that the first roll results in i and let Bj be the event that the second roll results in j. Are Ai and Bj independent? (b) Let A be the event that the first roll is a 1 and B be the event that the sum of the two rolls is a 5. Are A and B independent? (c) Let A be the event that the maximum of the two rolls is 2 and B be the event that the minimum of the two rolls is 2. Are A and B independent? 5. English and American spellings are valour and valor respectively. A man staying at a bed and breakfast writes this word, and a letter taken at random from his spelling is found to be a vowel. If 40 percent of the English-speaking men at the bed and breakfast are English and 60 percent are Americans, what is the probability that the person writing the word is an Englishman?

2 Homework Assignment 11w April 4, 2018

6. Consider an experiment involving two successive rolls of a fair die. Consider the fol- lowing events.

A: event that the first roll results in a 1. B: event that the second roll results in a 6. C: event that the sum of two rolls is a 7. D: event that the maximum of two rolls is 2. E: event that the minimum of two rolls is 5. Answer the following questions giving proper justification. (a) Are events A and B independent? (b) Are events A and C independent? (c) Are events B and C independent? (d) Are events A, B, and C independent? (e) Are events D and E independent?