1. Suppose we roll a fair k-sided die with the numbers 1 through k on the die’s
ID: 3703190 • Letter: 1
Question
1. Suppose we roll a fair k-sided die with the numbers 1 through k on the die’s 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?
Explanation / Answer
1. Suppose we roll a fair k-sided die with the numbers 1 through k on the die’s faces. If X is the number that appears, what is E[X]?
Answer:
The expectation is the sum of each value multiply its probability, so the equation should be as follows
1 * (1/k) + 2 * (1/k) + ... + k * (1/k) = rac{1}{k} * sum_{i=1}^{k}(i) = rac{k+1}{2}
So the result expectation E[X] = rac{k+1}{2}
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