CS206
Practice Problems II
April 13, 2012
(these are ALL practice problems, not to be handed in
1. Your computer program has 3 bugs. Each time you try some corrections there is a probability
of P = 1/3 that you remove some bug. Only one bug can be removed
Final Review
CS 206 Spring 2015
Prof. David Cash
1
Final Exam
Monday MAY 11 4PM-7PM
Makeup: Sunday MAY 10 12PM-3PM
Locations in Sakai announcement - multiple
rooms.
Same cheat-sheet rule
2
Preparing for the Final
Problems will be similar to
homew
CS 206 Practice Midterm II, Spring 2015
Prof. David Cash
Show your work on all problems unless otherwise directed.
You do not need to simplify answers.
Write your answers in the space provided in each problem. If you want an extra sheet
or back of a te
Rutgers University
CS206: Introduction to Discrete Structures II, Spring 2015
Professor David Cash
Homework 2 Solutions
1. (Suggested ungraded counting problems) In section 1.9 work problems 1, 3, 6, 8, 12.
2. (Suggested ungraded probability problems) In
CS 211: Computer Architecture, Spring 2012
Homework 2
Important: Make sure to write your name and section number on the HW. The latter will make it much
easier for the TAs to divide and grade the homework for
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CS 206 Practice Midterm I, Spring 2015
Prof. David Cash
Show your work on all problems unless otherwise directed.
You do not need to simplify answers.
Write your answers in the space provided in each problem. If you want an extra sheet
or back of
Rutgers University
CS206: Introduction to Discrete Structures II, Spring 2015
Professor David Cash
Homework 8
Due at the beginning of class on Friday, April 3
Instructions:
Exercises are labeled with point values.
Problems are worth 10 points each.
Thi
CS 206 Practice Midterm I, Spring 2015
Prof. David Cash
Show your work on all problems unless otherwise directed.
You do not need to simplify answers.
Write your answers in the space provided in each problem. If you want an extra sheet
or back of a tes
CS206
HW2
Feb. 5, 2012
Write up convincing answers to the questions marked by an asterisk (*). Some of the unstarred
questions will be worked in the recitations. (*) indicates a more challenging problem. It does
NOT have to be handed in, but if you do, we
Rutgers University
CS206: Introduction to Discrete Structures II, Spring 2015
Professor David Cash
Homework 10
Due at the beginning of class on Friday, April 24
Instructions:
There are no exercises this week. All problems are worth 15 points each.
As us
Rutgers University
CS206: Introduction to Discrete Structures II, Spring 2015
Professor David Cash
Homework 11
Due at the beginning of class on Friday, May 1
Instructions:
Problems are worth 10 points each. There are no exercises this week.
As usual, Wr
Rutgers University
CS206: Introduction to Discrete Structures II, Spring 2015
Professor David Cash
Homework 3
Due at the beginning of class on Friday, Feb 13
Instructions: All problems on this homework are worth 10 points. Write your solutions neatly or
t
CS206
HW4
(* due date is April 2, 2012)
March 21, 2012
1. (*) A fair die is tossed twice. Let X = the sum of the faces, Y = the maximum of the two
faces, and Z = |face 1 face 2|.
(a) Write down the value of X ,Y , Z , and W = XZ for each outcome w S .
(b)
Rutgers University
CS206: Introduction to Discrete Structures II, Spring 2015
Professor David Cash
Homework 7
Due at the beginning of class on Friday, March 27
Instructions:
Exercises are worth 5 points each.
Problems are worth 10 points each.
This hom
Rutgers University
CS206: Introduction to Discrete Structures II, Spring 2015
Professor David Cash
Homework 6
Due at the beginning of class on Friday, March 13
Instructions:
Exercises are worth 5 points each.
Problems are worth 10 points each.
This hom
CS 206 Quiz 1 Practice, Spring 2015
Name:
Please do not write in this space.
RUID:
/5 +
/5 = Total:
/10
1. Consider the following experiment: We put three slips of paper labeled 1, 2, 3 into a
hat, and then draw 2 slips at random, with replacement. Let A
Rutgers University
CS206: Introduction to Discrete Structures II, Spring 2015
Professor David Cash
Homework 1 Solutions
1. Suppose A, B, C are sets in a some universe U , and that |A| = 5, |B| = 8, |C| = 4, |U | = 10.
(Im not saying which sets A, B, C, U
Rutgers University
CS206: Introduction to Discrete Structures II, Spring 2015
Professor David Cash
Homework 2
Due at the beginning of class on Friday, Feb 6
Instructions: All problems on this homework are worth 10 points. Write your solutions neatly or
ty
Jointly Distributed
Random Variables
Reading:
Ross, Ch 6.
04/20/2009
CS206 - Intro. to Discrete
Structures II
1
Example 1
Consider an urn with 3 red, 4 white and 5 black
balls. Draw three balls randomly from the urn.
What is the probability that there wi
Conditional Probability
Reading:
Ross, Ch 3., Sec. 1-2 & 5.
02/23/2009
CS206 - Intro. to Discrete
Structures II
1
Example
Two dice, 36 equally likely outcomes.
E1: first dice is 3.
E2: sum of two dice = 6.
P(E1)= ? P(E2) = ? P(E2 and E1) = ?, P(E2 if E1
Expectation & Variance
Reading:
Ross, Ch 4., Sec. 3-5.
03/22/2009
CS206 - Intro. to Discrete
Structures II
1
3 tosses of a coin
Recall the problem of tossing a coin three times and the
profit the player earns from this game:
Outcome
P(O)
Profit
HHH
1/8
3
Probability II:
Axioms & Properties
Reading:
Ross, Ch 2., Sec. 3-5 & 7
02/16/2009
CS206 - Intro. to Discrete
Structures II
1
Recall: Probability of Events
If the sample space of an experiment is finite
and if all atomic events are equally likely, then t
Normal (Gaussian)
Random Variables
Reading:
Ross, Ch 5., Sec. 4.
Ch. 8
04/15/2009
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-1p
Draw 1001000 sa
Probability I:
Basic Ideas
Reading:
Ross, Ch 2., Sec. 1.
02/16/2009
CS206 - Intro. to Discrete
Structures II
1
Intuitive notion of probability
What is the probability of heads when tossing a fair coin?
Two outcomes, heads and tails.
Both equally likely.
Transformations of
Random Variables
Reading:
Ross, Ch 5, Sec. 7. Ch. 6, Sec. 7
04/15/2009
Thursday, December 3, 2009
CS206 - Intro. to Discrete
Structures II
1
Functions of RVs
Suppose you take a deterministic transformation
(function) of random variable
Continuous Random
Variables
Reading:
Ross, Ch 5., Sec. 1-2.
04/01/2009
CS206 - Intro. to Discrete
Structures II
1
Continuous Random Variable
X is a continuous random variable if it takes on
values from the (sub)set of real numbers:
X 2 <.
Examples:
Pos
Simulation of R.V.s
Reading:
Ross, Ch 10, Sec. 1-4
04/01/2009
CS206 - Intro. to Discrete
Structures II
1
How to generate samples of rvs?
We typically start with some random experiment and
seek to describe the pmf of a rv in that experiment
But what if w
Jointly Distributed
Random Variables
Reading:
Ross, Ch 6.
04/20/2009
CS206 - Intro. to Discrete
Structures II
1
Example 1
Consider an urn with 3 red, 4 white and 5 black
balls. Draw three balls randomly from the urn.
What is the probability that there wi
Expectation & Variance
Reading:
Ross, Ch 4., Sec. 3-5.
03/22/2009
CS206 - Intro. to Discrete
Structures II
1
3 tosses of a coin
Recall the problem of tossing a coin three times and the
profit the player earns from this game:
Outcome
P(O)
Profit
HHH
1/8
3
Probability II:
Axioms & Properties
Reading:
Ross, Ch 2., Sec. 3-5 & 7
02/16/2009
CS206 - Intro. to Discrete
Structures II
1
Recall: Probability of Events
If the sample space of an experiment is finite
and if all atomic events are equally likely, then t
Rutgers University
CS206: Introduction to Discrete Structures II, Summer 2016
Homework 2
Instructions
Please staple a blank page containing your name and RUID at the front of the assignment.
Your submission MUST be stapled, loose pages will not be accept
Problem 1
This problem consists of four unrelated counting problems. You do not need to explain your
answers in this problem.
(a) How many ways are there to arrange the letters in TENNESSEE?
Gl l
"' .OK
4! 2. U 4 3L 1
(b) What is the coefficient of cfw_89
Rutgers University
CS206: Introduction to Discrete Structures II, Summer 2016
Homework 3
Instructions
Please staple a blank page containing your name and RUID at the front of the assignment.
Your submission MUST be stapled, loose pages will not be accept
Rutgers University
CS206: Introduction to Discrete Structures II, Summer 2016
Homework 6
Instructions
Please staple a blank page containing your name and RUID at the front of the assignment.
Your submission MUST be stapled, loose pages will not be accept
CS206: Introduction to Discrete Structures II
Tail Inequalities
1
Tail Inequalities
The purpose of tail inequalities is to bound how far away a random variable can be from its expected value.
We study two ways to estimate the tail probabilities of random