Mathematical Foundations of Computer Science Homework Assignment 12w

Mathematical Foundations of Computer Science
Homework Assignment 12w
Given: November 12, 2014
Due: November 17, 2014
This assignment is due at the beginning of the class on the due date. Unless all problems
carry equal weight, the point value of each problem is shown in [ ]. To receive full credit all
your answers should be carefully justified. You can work in groups of upto three students
from this class. Each solution must be written independently by each student. You must
list the students that you worked with on this homework.
1. Suppose that 90% of web merchants that sell light bulbs are sleazy: they’ll sell you a
light bulb that has a 1/20 chance of burning out on any day. The remainder 10% of web
merchants use high-quality merchandise: their bulbs have a 1/1000 chance of burning out
on any given day. Sadly, you can’t distinguish the two types of merchants apart from their
web site – they all look identical.
I pick a web merchant at random, buy a dozen bulbs from them, and install the first
bulb in my kitchen. Let the random variable X denote the number of days until it burns
out. When it burns out, I curse loudly, then replace it with another bulb from the same
merchant (because I’ve got to use them up). Let the random variable Y denote the number
of days until the second bulb burns out (counting from the day when we replaced the first
bulb).
Are the random variables X and Y independent?
2. Eight men and seven women are randomly assigned distinct numbers from 1 to 15. On
an average, how many pairs of consecutive numbers are assigned to people of opposite sex?
3. Here are seven propositions:
x1 ∨ x2 ∨ x7
x5 ∨ x6 ∨ x7
x2 ∨ x4 ∨ x6
x4 ∨ x5 ∨ x7
x3 ∨ x5 ∨ x8
x9 ∨ x8 ∨ x2
x3 ∨ x9 ∨ x4
Note that
1. Each proposition is the OR of the three terms of the form xi or of the form xi .
2. The variables in the three terms in each proposition are all different.
Suppose that we assign true/false values to the variables x1 , . . . , x9 independently and with
equal probability.
2
Homework Assignment 12w
November 12, 2014
a. What is the probability that a single proposition is true?
b. What is the expected number of true propositions?
c. Use your answer to prove that there exists an assignment to the variables
that makes all of the propositions true.
4. If G is a connected graph in which every vertex has even degree then G has no edge
whose deletion leaves a disconnected graph.
5. An Eulerian walk in a graph is a walk that traverses each edge exactly once and
returns to a different vertex than where it started from. Prove that a connected graph has
an Eulerian walk if and only if it has exactly two vertices of odd degree.