EECS 70
Fall 2014
Discrete Mathematics and Probability Theory
Anant Sahai
Discussion
2M
1. Two Color Theorem
Consider a scenario where we have a 2D plane that we divide into regions by drawing straight lines. Using
induction, prove that we can color this
EECS 70
Discrete Mathematics and Probability Theory
Spring 2014
Anant Sahai
Homework 4
This homework is due Feb 17 2014, at 12:00 noon.
1. Pentagons, Pentagrams, and Pythagoreans: a high-school geometry proof of the existence of irrational
numbers by way
EECS 70
Discrete Mathematics and Probability Theory
Spring 2014
Anant Sahai
Homework 9
This homework is due Mar 31 2014, at 12:00 noon.
1. Virtual Lab 3: Biased Coins Continued
In this problem we will continue the lab from last HW. We will start from the
EECS 70
Discrete Mathematics and Probability Theory
Spring 2014
Anant Sahai
Homework 2
This homework is due Feb 3 2014, at 12:00 noon.
1. Matrix Induction
1 1
Guess a formula for the matrix computation
0 1
n
and prove by induction that your formula is cor
EECS 70
Discrete Mathematics and Probability Theory
Fall 2014
Anant Sahai
Homework 10
This homework is due November 10, 2014, at 12:00 noon.
1. Section Rollcall!
In your self-grading for this question, give yourself a 10, and write down what you wrote for
EECS 70
Discrete Mathematics and Probability Theory
Fall 2014
Anant Sahai
Homework 8
This homework is due October 27, 2014, at 12:00 noon.
1. Section Rollcall!
In your self-grading for this question, give yourself a 10, and write down what you wrote for p
EECS 70
Discrete Mathematics and Probability Theory
Spring 2014
Anant Sahai
Homework 7
This homework is due Mar 10 2014, at 12:00 noon.
1. Programming Question and Virtual Lab: Intro to Randomness
This problem is a simple exercise in programming that is d
EECS 70
Discrete Mathematics and Probability Theory
Spring 2014
Anant Sahai
Homework 13
This homework is due April 28 2014, at 12:00 noon.
1. Von Neumanns Unbiaising Method
In 1951, John Von Neumann gave the following simple procedure to simulate a perfec
EECS 70
Discrete Mathematics and Probability Theory
Fall 2014
Anant Sahai
Homework 7
This homework is due October 20, 2014, at 12:00 noon.
1. Section rollcall!
In your self-grading for this question, give yourself a 10, and write down what you wrote for p
EECS 70
Fall 2014
Discrete Mathematics and Probability Theory
Anant Sahai
Discussion
8M
1. Corruption
Alice wants to send Bob a message of size 3 and guard against 1 corrupt packet. Bob receives from Alice
the packets (0, 1), (1, 2), (2, 3), (3, 3), (4, 4
EECS 70
Discrete Mathematics and Probability Theory
Fall 2014
Anant Sahai
Homework 8
This homework is due October 27, 2014, at 12:00 noon.
1. Section Rollcall!
In your self-grading for this question, give yourself a 10, and write down what you wrote for p
EECS 70
Discrete Mathematics and Probability Theory
Spring 2014
Anant Sahai
Homework 11
This homework is due Apr 14 2014, at 12:00 noon.
1. Virtual Lab
The Birthday Paradox concerns the probability of two people in a group of m people having the same birt
EECS 70
Fall 2014
Discrete Mathematics and Probability Theory
Anant Sahai
Discussion
12W
1. Cookies Again
GSIs still have plenty of cookies left from last section, so we are giving out cookies again! To make it more
interesting, we give exactly 10 cookies
EECS 70
Discrete Mathematics and Probability Theory
Spring 2014
Anant Sahai
Homework 10
This homework is due Apr 7 2014, at 12:00 noon.
1. Maze
Lets assume that Tom is located at the bottom left corner of the maze below, and Jerry is located at the top ri
EECS 70
Discrete Mathematics and Probability Theory
Spring 2014
Anant Sahai
Homework 14
This homework is due May 05 2014, at 12:00 noon.
1. Tolerating Errors
Assume Alice is trying to send m packets across a noisy channel to her friend Bob. The channel in
EECS 70
Discrete Mathematics and Probability Theory
Fall 2014
Anant Sahai
Homework 7
This homework is due October 20, 2014, at 12:00 noon.
1. Section rollcall!
In your self-grading for this question, give yourself a 10, and write down what you wrote for p
EECS 70
Discrete Mathematics and Probability Theory
Fall 2014
Anant Sahai
Homework 12
This homework is due November 24, 2014, at 12:00 noon.
1. Section Rollcall!
In your self-grading for this question, give yourself a 10, and write down what you wrote for
EECS 70
Fall 2014
Discrete Mathematics and Probability Theory
Anant Sahai
Discussion
10W
1. Story Problems
Prove the following identities by combinatorial argument:
(a)
n
n1
n1
=
+
k
k1
k
(b)
2n
n
=2
+ n2 .
2
2
(c)
n
k
k=0
(d)
n
k= j
EECS 70, Fall 2014, D
EECS 70
Fall 2014
Discrete Mathematics and Probability Theory
Anant Sahai
Discussion
12M
1. Lets give you cookies!
The GSIs want to distribute k cookies and k glasses of milk to students. They will blindly pick a student and
give her/him a cookie. Same go
EECS 70
Discrete Mathematics and Probability Theory
Fall 2014
Anant Sahai
Homework 10
This homework is due November 10, 2014, at 12:00 noon.
1. Section Rollcall!
In your self-grading for this question, give yourself a 10, and write down what you wrote for
EECS 70
Discrete Mathematics and Probability Theory
Fall 2014
Anant Sahai
Homework 11
This homework is due November 17, 2014, at 12:00 noon.
1. Section Rollcall!
In your self-grading for this question, give yourself a 10, and write down what you wrote for
EECS 70
Fall 2014
Discrete Mathematics and Probability Theory
Anant Sahai
Discussion
4W-S
1. Euclids Algorithm Euclids algorithm is a fast algorithm for computing the greatest common divisor of
two integers. Here is an example. To compute gcd(16, 10):
16
EECS 70
Fall 2014
Discrete Mathematics and Probability Theory
Anant Sahai
Note 15
Random Variables: Distributions, Independence, and Expectations
In the last note, we saw how useful it is to have a way of thinking about quantities that are inherently rand
EECS 70
Fall 2014
Discrete Mathematics and Probability Theory
Anant Sahai
Discussion
9M
1. Probability Practice
(a) A message source M of a digital communication system outputs a word of length 8 characters, with the
characters drawn from the ternary alph
EECS 70
Discrete Mathematics and Probability Theory
Fall 2014
Anant Sahai
Homework 12
This homework is due November 24, 2014, at 12:00 noon.
1. Section Rollcall!
In your self-grading for this question, give yourself a 10, and write down what you wrote for
EECS 70
Discrete Mathematics and Probability Theory
Fall 2014
Anant Sahai
Homework 7
This homework is due October 20, 2014, at 12:00 noon.
1. Section rollcall!
In your self-grading for this question, give yourself a 10, and write down what you wrote for p
EECS 70
Fall 2014
Discrete Mathematics and Probability Theory
Anant Sahai
Discussion
9W
1. Balls and Bins
You have n empty bins and you throw balls into them one by one randomly. A collision is when a ball is
thrown into a bin which already has another ba
EECS 70
Discrete Mathematics and Probability Theory
Fall 2014
Anant Sahai
Homework 6
This homework is due October 13, 2014, at 12:00 noon.
1. Try It Out (optional)
Please do the online problems on FLT and RSA. Give us your brief comments on how you found
EECS 70
Fall 2014
Discrete Mathematics and Probability Theory
Anant Sahai
Discussion
3M-S
1. Stable Matching Say that we want to pair up the men (1,2,3) with the women (A,B,C); such a pairing is
called a matching. The table below gives the ranked preferen
EECS 70
Fall 2014
Discrete Mathematics and Probability Theory
Anant Sahai
Discussion
1. Chebyshevs Inequality
Derive the Chebyshevs Inequality Pr[|X | ]
Var(X)
13M
[X]
using Markov Inequality Pr[X ] E .
2. Working with the law of large numbers
(a) A fair
Today
Finish Euclid.
Bijection/CRT/Isomorphism.
Review for Midterm.
Finding an inverse?
We showed how to efficiently tell if there is an inverse.
Extend euclid to find inverse.
Euclids GCD algorithm.
(define (euclid x y)
(if (= y 0)
x
(euclid y (mod x y)
Probability
Probability!
Confuses us. But really neat.
At times, continuous. At others, discrete.
Sample Space:, Pr [].
Random Variable: X
Event: Pr [A] = A Pr []
Event: A = [a, b], Pr [X A],
Pr [] = 1.
CDF: F (x) = Pr [X x].
Random variables: X ().
PDF:
\Questioncfw_Error-Correcting Codes
\begincfw_enumerate
\renewcommandcfw_\labelenumicfw_(\alphcfw_enumi)
\item
Recall from class the error-correcting code for erasure errors, which
protects against up to $k$ lost packets by sending a total of $n+k$ packet
CS 70
Discrete Mathematics and Probability Theory
Spring 2017
Rao
1
DIS 13b
Continuous Computations
Let X be a continuous random variable whose pdf is cx3 (for some constant c) in the range 0 x
1, and is 0 outside this range.
(a) Find c.
(b) Find Pr[1/3
CS 70
Spring 2017
1
Discrete Mathematics and Probability Theory
Rao
DIS 13a
Uniform Means
Let X1 , X2 , . . . , Xn be n independent and identically distributed uniform random variables on the
interval [0, 1].
(a) Let Y = mincfw_X1 , X2 , . . . , Xn . Find