Rutgers University
CS206: Introduction to Discrete Structures II, Spring 2016
Solutions by Abdul Basit
Homework 6 Solutions
1. The parts of this problem are unrelated.
(a) (5 points) We deal a five card hand. Let A be the event that all 5 cards are the sa

Rutgers University
CS206: Introduction to Discrete Structures II, Spring 2016
Solutions prepared by Hai Xuan Pham
Homework 10 - Solutions
1. For each of the statements below, determine if they are true or false. If it is true, explain
why, using any resul

Rutgers University
CS206: Introduction to Discrete Structures II, Spring 2016
Solutions prepared by Hai Xuan Pham
Homework 7 Solutions
1. (10 points) Suppose we play a modified version of the Monty Hall game. In this version, there
are 5 doors instead ins

Rutgers University
CS206: Introduction to Discrete Structures II, Spring 2016
Professor David Cash
Homework 9 Solutions
1. (5 points) Fix two distinct numbers a, b, and let X be a random variable such that P (X =
a) = p and P (X = b) = 1 p. Find E(X) and

Rutgers University
CS206: Introduction to Discrete Structures II, Spring 2016
Professor David Cash
Homework 3 - Solutions
1. To fulfill the requirements for a degree, a student can choose to take any 5 out of 25 courses,
with the constraint that at least

Rutgers University
CS206: Introduction to Discrete Structures II, Spring 2016
Professor David Cash
Homework 4 Solutions
1. We are going to divide a collection of 20 children into groups. The order of the teams does
not matter (i.e. the teams are unnamed).

Rutgers University
CS206: Introduction to Discrete Structures II, Spring 2016
Professor David Cash
Homework 8 Solutions
1. (5 points) We roll two modified six-sided die that have faces 1, 1, 2, 3, 4, 4, all shown with
equal probability. Let X be the face

A Sample Solution for Homework 5
1 All the workers at a certain company drive to work and park in the
companys lot. The company is interested in estimating the average
number of workers in a car. Which of the following methods will
enable the company to e

CS206: Practice Problem Solutions
Set Theory and Axioms of Probability
1.
(a) E F c Gc
(b) E G F c
(c) E F G
(d) (E F ) (E G) (F G)
(e) E F G
(f) E c F c Gc
(g) S \ (E F ) (E G) (F G)
(h) S \ (E F G)
(i) (E F Gc ) (E F c G) (E c F G)
(j) E F G (E c F c Gc

Bayes Rule
Reading:
Ross, Ch 3., Sec. 3.
02/25/2009
CS206 - Intro. to Discrete
Structures II
1
Complementary causes
Let E and C be two events . Then:
P(E) = P(E|C)P(C) + P(E|Cc)P(Cc)
Cc
C
10/20/2015
E
CS206 - Intro. to Discrete Structures II
2
Example: t

Independence
Reading:
Ross, Ch 3., Sec. 4.
02/25/2009
CS206 - Intro. to Discrete
Structures II
1
Example 1
From a deck of 52 cards we draw one card randomly. What is the
probability that this card will be an ace?
(a) 1/13
(b) C(52,4)/C(52,1) (c) 1/4
10/

CS206: Introduction to Discrete Structures II
Questions
Distributions, Expected Value and Variance
1. Bob is playing a video game that has 7 levels. He starts at level 1, and has probability p1 of reaching
level 2. In general, given that he reaches level

CS206: Introduction to Discrete Structures II
Probability Distributions
Bernoulli
An experiment that can result in either a success or a failure (but not both) is called a Bernoulli trial.
A Bernoulli random variable can be thought of as the indicator of

CS206: Introduction to Discrete Structures II
Final Exam
Date: July 16, 2015
Instructions
You may use ONE page of prepared notes (both sides), but otherwise the test is closed book. All work
must be your own.
Show ALL your work. You will get little or n

CS206: Midterm Solutions
1. The size of the sample space is 65 .
(a) There are 65 ways to pick 5 numbers, and they can appear in 5! different ways.
6
5 5!
P (no two alike) = 5
6
(b) There are 61 ways to pick one number that appears twice and 52 ways to pi

CS206: Practice Problems
Set Theory and Axioms of Probability
1. Let E, F , and G be three events. Find expressions for the events so that, of E, F , and G,
(a) only E occurs;
(b) both E and G, but not F , occur;
(c) at least one of the events occurs;
(d)

CS206: Introduction to Discrete Structures II
Homework 4 Solution
1
Question 1
Let X be the number of children needed, starting with the 2nd child, to obtain one whose gender is not the same as
that of the firstborn. Then the distribution of Y = X 1 is th

Unordered Sampling
with Replacement
Reading:
Ross, Ch 1., Sec. 5.
Rosen, Ch 5., Sec. 5.
02/02/2009
CS206 - Intro. to Discrete
Structures II
1
Example 1
How many ways to pick four pieces of fruit from a bowl
of five different oranges?
(a) C(5,4) (b) P(5,4

Pigeonhole Principle
Reading:
Ross, Ch 1., Sec. 1.
Rosen, Ch 6., Sec. 2
01/21/2009
CS206 - Intro. to Discrete
Structures II
1
Objects and Boxes
Suppose there are 13 pigeons in a flock and 12
pigeonholes for them to roost. Will all pigeons be able to
find

Permutations &
Combinations
Reading:
Ross, Ch 1., Sec. 3 and 4.
Rosen, Ch 6., Sec. 3
02/02/2009
CS206 - Intro. to Discrete
Structures II
Adapted from Detlef Ronneburger
1
Words from letters
How many 7-letter words (combinations of letters) can be
formed

Rutgers University
CS206: Introduction to Discrete Structures II, Spring 2016
Solutions by Hai Xuan Pham
Homework 5 - solutions
Instructions: All problems on this homework are worth 10 points, including the extra credit
question.
1. (10 points) Give a sto

Rutgers University
CS206: Introduction to Discrete Structures II, Spring 2015
Professor David Cash
Homework 2 Solutions
1. (25 points) In all parts of this question, ID numbers are 5 digits long (where each digit comes
from 0 through 9).
(a) How many poss

Rutgers University
CS206: Introduction to Discrete Structures II, Spring 2016
Homework 1 Solutions
1. Suppose E, F, G are events in a some sample space S. In class, we learned that, for instance,
E F can be interpreted as E implies F . For each of the rel

Stirling Numbers
Reading:
Rosen, Ch. 6 & Notes
02/02/2009
CS206 - Intro. to Discrete
Structures II
1
Balls & Urns: Occupancy Problems
Many problems in combinatorics can be cast as
placing balls into urns/cells.
How do we distinguish different types of s

Principles of
Counting
Reading:
Ross, Ch 1., Sec. 1.
Rosen, Ch 5., Sec. 1
01/21/2009
CS206 - Intro. to Discrete
Structures II
Adapted from Detlef Ronneburger
1
Counting Poker Hands
Poker is played with 52 cards
13 faces (2,3,4,5,6,7,8,9,10,J,Q,K,A) in f

Discrete Structures II :
Introduction
Reading for next meeting:
Ross, Ch 1., Sec. 1.
Rosen, Ch 5., Sec. 1
01/21/2009
CS206 - Intro. to Discrete
Structures II
1
Today
Class logistics
What is Discrete Structures II about?
Why you should know about Discre

A Sample Solution for Homework 6
1 Each night dierent meteorologists give us the probability that it will
rain the next day. To judge how well these people predict, we will
score each of them as follows: If a meteorologist says that it will rain
with prob

Normal (Gaussian)
Random Variables
Reading:
Ross, Ch 5., Sec. 4.
04/15/2009
Thursday, April 23, 2009
CS206 - Intro. to Discrete
Structures II
1
Example 1
Consider a discrete random variable X, with geometric distribution, X ~
Geom(p=0.1).
P(X=i) = (1-p)i-

CS206: Introduction to Discrete Structures II
Homework 2
Due Thursday, July 9, 2015
1. We have n boxes and k balls. We want to determine the number of ways in which the balls can be
distributed among the boxes. We consider variants the following: balls ar

CS206: Introduction to Discrete Structures II
Probabilistic Method
1
The Probabilistic Method
The probabilistic method is a remarkable technique for proving the existence of combinatorial
objects with specified properties. It is based on probability theor

CS206: Introduction to Discrete Structures II
Homework 1
Due Thursday, July 2, 2015
1. Let A, B, A1 , A2 . . . , An be sets in some universe S. Prove the following:
(a) (A B)c = (Ac ) (B c )
S
T
(b) ( ni=1 Ai )c = ni=1 (Ai )c . Use part (a) and induction

CS206: Introduction to Discrete Structures II
Midterm Exam
Date: July 16, 2015
Instructions
You may use ONE page of prepared notes (both sides), but otherwise the test is closed book. All work
must be your own.
Show ALL your work. You will get little or

CS206: Practice Problems 2 Solutions
Distributions, Expected Value and Variance
1. Let X be the highest level Bob reaches. Then X can be viewed as a generalization of the First Success.
The probability that he reaches level 2 is p1 , level 3 is p1 p2 . In