CS206
Apr. 15, 2014
HW4 - Some Solutions
Question 2: I will just do X and Z. Lets dene a matrix for X using X(i, j) = i + j, where
i, j = 1, . . . , 6. The entries in ROW i give the sum of the faces when the rst die shows i and the
entries in COL. j show
CS206
HW1
Sept. 8, 2009
(Write up convincing answers to the questions marked by an asterisk (*). Some of the unstarred questions will be worked in recitation. (*) indicates a more challenging problem, not required to hand in unless you want feedback - its
CS206
Review Sheet 4
April 13, 2014
1. Variance: Suppose X is a random variable on a probability space (S, P ) and with expected value E(X) = m. The variance of X is the expected squared deviation from m,
dened by
V (X) = E([X m]2 ).
(1)
Evaluating (1) ov
CS206
Review Sheet 5
April 26, 2014
1. The Random Walk on the Integers: We start with the coin toss game which uses a
coin with probability P for Head and probability q = 1 P for Tail. You get a dollar if it
comes up Head and lose a dollar it it shows Tai
CS206
OLD TEST
SOME SOLS.
1a: 2(k!)(k!)/(2k)!. There are (2k)! ways to order the n = 2k integers. If the odds and evens
alternate, there are k! ways to order EACH of these subsets and 2 ways to alternate them,
ONE if an odd integer is rst and the OTHER i
CS206
HW4 (* due date is November 4, 2009)
October 20, 2009
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)
CS206
March 12, 2014
TEST 1
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 no credit for an unexplained answer.
Do all your
CS206
October 27, 2010
TEST 1
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 no credit for an unexplained answer.
Do all yo
CS206
December 19, 2008
FINAL TEST
Instructions:
Do all your work in the blue exam books.
You may use two pages of prepared formulas and notes, but all work must be
your own.
Please write your answers IN THE GIVEN ORDER, though you may solve problems i
CS 206 Quiz 4 Solutions, Summer 2016
Name: Solutions
RUID:
Please do not write in this space.
Total:
/10
1. Oscar has lost his dog in either forest A (with probability 0.3) or in forest B (with
probability 0.2) or forest C (with probability 0.5). If the d
So the solution is: (4 x 5 x 6)/(4 x 6 x 8)For Practice Midterm 1:
Question 2(d) The solutions say: (4 x 5 x 7)/(4 x 6 x 8). This is wrong!
The correct solution is the following:
Pick a number for the 4 face die. There are 4 ways to do this.
The number of
CS 206 Quiz 5 Solutions, Summer 2016
Name: Solutions
Please do not write in this space.
RUID:
/2 +
/4 +
/4 = Total:
/10
1. Let X Unif(3), i.e. Range(X) = cfw_1, 2, 3.
(a) What is the PMF of X? Draw the PMF.
(b) Let Y = |X 2|. What is the Range and PMF of
Rutgers University
CS206: Introduction to Discrete Structures II, Summer 2016
Recitation 10 Solutions
1. We toss 3 balls into 3 bins, with all outcomes equally likely, and let N be the number of
empty bins. Find E(N ) and Var(N ).
Solution: We can use ind
Rutgers University
CS206: Introduction to Discrete Structures II, Summer 2016
Recitation 7 Solutions
1. This problem asks you find the distributions of some random variables. Recall that this
means you should find the probabilities P (X = k) for each k, e
Rutgers University
CS206: Introduction to Discrete Structures II, Summer 2016
Recitation 12 Solutions
1. Each game you play is a win with probability 0.5. You plan to play 5 games, but if you win
the fifth game, then you will keep on playing until you los
Rutgers University
CS206: Introduction to Discrete Structures II, Summer 2016
Homework 5 Solutions
1. Let X be uniformly distributed on the set C = cfw_2, 1, 0, 1, 2. How would you write X as a
a fuction of a Uniform Random Variable? Draw the graph of X.
Question 2(c) The question paper and the solutions have 2 different problems. In the question paper, it says:
"Now Walter insists that he eats at his absolute favorite restaurant, Los Pollos Hermanos, at least twice that week."
In the solutions, it says:
CS 206 Final, Spring 2015 (Form B)
Instructor: David Cash
TAs: Amr Bakry, F atma Durak, Hai Pham
Undergraduate graders: Eric Brugel7 Ryan Dunn
You may use a one—sided7 handwritten “cheat sheet.” A formula sheet is provided with
the exam. N 0 other resourc
Rutgers University
CS206: Introduction to Discrete Structures II, Summer 2016
Homework 4 Solutions
1. (a) We flip a fair coin 3 times. Let A be the event that Heads comes up an odd number of
times, and let B be the event that the first toss is Heads. Are
Rutgers University
CS206: Introduction to Discrete Structures II, Summer 2016
Homework 3 Solutions
1. What is the probability that a 5-card poker hand has exactly three spades? What is the
probability that it has at least three spades?
Solution: We count
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?
(b) What is the coefficient of x9 y 4 when (x + y)13 is expanded
CS 206 Quiz 2 Solutions, Summer 2016
Name: Solutions
RUID:
Please do not write in this space.
Total:
/10
1. We are given FOUR coins: one has heads on both faces, the second has tails on both
faces, and the other two have heads on one face and a tail on th
Rutgers University
CS206: Introduction to Discrete Structures II, Summer 2016
Recitation 11 Solutions
1. Find the expected number of face cards in a 5 card poker hand. (A face card is a Jack, Queen,
or King.)
Solution: Let X be the number of face cards in
Problem 1
The parts of this problem are unrelated.
(a) How many ways are there to arrange the letters of MISSISSIPPI? (Swapping identical
letters does not count as a new arrangement.)
(b) How many ways are there to distribute 10 balls into 5 baskets, if t
CS 206 Practice Final, Spring 2015
Prof. David Cash
You may use a one-sided, handwritten cheat sheet. A formula sheet is provided with
the exam. No other resources are allowed.
Write your answers in the space provided in each problem. An extra blank she
CS 206 Quiz 1 Solutions, Summer 2016
Name: Solutions
RUID:
Please do not write in this space.
/5 +
/5 = Total:
/10
1. Prove using the axioms of probability that
P (E F c ) = P (E) P (E F ).
Solutions:
Note that we can write F = (F E c ) (F E). Now since t
Rutgers University
CS206: Introduction to Discrete Structures II, Summer 2016
Homework 2 Solutions
1. From a group of 3 freshmen, 4 sophomores, 4 juniors, and 3 seniors a committee of size 4 is
randomly selected. We assume that all people are distinguisha
Chapter 3
Problems
1.
Pcfw_6 different = Pcfw_6, different/Pcfw_different
Pcfw_1st = 6,2nd 6 + Pcfw_1st 6,2nd = 6
=
5/6
2 1/ 6 5 / 6
=
= 1/3
5.6
could also have been solved by using reduced sample spacefor given that outcomes differ it
is the same as aski
Graded Exercises
1. P(A) = 4/8
P(B) =
P(A B) = 2/8
4/9 * = 2/8
They are independent because B does not affect the probability of A since it is heads.
2.P(C) = 4/8
P(A) = 4/8
P(A C) = 2/8
4/8 * 4/8 = 2/8
They are independent because C does not affect the
Homework 1
1) (a) 5 because if (B C^c) does not intersect with A, then the size of the set difference would be
|A| which is 5.
(b) (B C^c) can have max size of 8 if no members of C are in B. 8 is bigger than 5, the size of A, so
if A is a subset of (B C^c