Handout D Challenges to Solving Probability Problems
There are many things to think about when solving a probability problem, including: Establish the UniverseSample Space Establish outcomes ( experiments ! sample space) Establish the events in the
Introduction to Probability and Statistics for Electrical Engineering
EE 364

Fall 2014
Example 1
In your summer internship, you are working for the worlds largest producer of light bulbs.
Your manager asks you to estimate the quality of production, that is, to estimate the
probability that a bulb produced by the factory is defectless. You
Introduction to Probability and Statistics for Electrical Engineering
EE 364

Fall 2014
EE364 FALL 2014: M IDTERM S OLUTIONS
Oct. 9, 2014
P ROBLEM 1 (5+5+5+15+10=40 pts): Circle the correct answer below
(a) Is the following statement TRUE or FALSE?
P (A1 A2 A3 An1 An )
= P (A1 ) P (A2 A1 ) P (A3 A1 A2 ) P (An A1 A2 A3 An1 )
(b) A wellshu
J. M. Mendel Spring 2010
EE 364 Midterm # 2
Closed Book Two 81/2X11 sheet of notes (both sides) permitted Calculator required Tables A.1, A.2 and A.3 required To receive full credit you must carry out all of the numerics
x
1. (20 points) For parts a and b
Introduction to Probability and Statistics for Electrical Engineering
EE 364

Spring 2016
EE 364 Homework 3
In Walpole et al., do the following exercises:
Sec 2.5 (pp. 5962): 2.72
Sec 2.6 (pp. 6971): 2.74, 2.82, 2.86, 2.88, 2.92
Sec 2.7 (pp. 7677): 2.96, 2.98 (assume that the events of passing through the four
locations are mutually exclusi
Introduction to Probability and Statistics for Electrical Engineering
EE 364

Spring 2016
EE 364 Homework 4
In Walpole et al., do the following exercises:
Sec 2.6 (pp. 6971): 2.76, 2.94
Sec 2.7 (pp. 7677): 2.102 (Assume that the host knows which door you picked and
which door the prize was behind, and always opens a door that you didnt pick
Introduction to Probability and Statistics for Electrical Engineering
EE 364

Spring 2016
EE 364 Homework 1
In Walpole et al., do the following exercises:
Sec 2.2 (pp. 4244): 2.2, 2.6, 2.12, 2.16, 2.18
Sec 2.3 (pp. 5152): 2.28, 2.32, 2.36
In addition, do the problems below:
Problem 1: Suppose you flip a coin repeatedly until you get heads. Y
Introduction to Probability and Statistics for Electrical Engineering
EE 364

Spring 2016
Practice Problem Set 2
1. A box contains 3 blue and 2 red marbles while another box contains 2 blue and 5 red
marbles. A marble drawn at random from one of the boxes turns out to be blue. What is the
probability that it came from the first box?
2. A poker
Introduction to Probability and Statistics for Electrical Engineering
EE 364

Spring 2016
Practice Problem Set 1
1. Consider the experiment of tossing a fair coin repeatedly and counting the number of tosses
required until the first head appears.
a. Find the sample space of the experiment.
b. Find the probability that the first head appears on
Introduction to Probability and Statistics for Electrical Engineering
EE 364

Spring 2016
EE 364 Homework 2
In Walpole et al., do the following exercises:
Sec 2.3 (pp. 5152): 2.38, 2.42, 2.48
Sec 2.5 (pp. 5962): 2.52, 2.56, 2.58, 2.64, 2.66
In addition, do the problems below:
Problem 1: A poker hand consists of 5 cards drawn from a wellshuf
Introduction to Probability and Statistics for Electrical Engineering
EE 364

Spring 2016
Lecture 1: Introduction
1. Use of words that convey uncertainty in our daily language.
a. We say probable, likely, plausible, credible, impossible, presumptive,
improbable, etc. We say have a good chance, there is a reasonable prospect,
it is highly likel
EE364 FALL 2014: T UTORIAL #11
Monday, Nov. 17, 2014
Note that for Homework 5 (and for this tutorial) the tables in the back of the book dont always give
exact numbers. Approximate to the best of your ability.
P ROBLEM 1 (Rapidre review): Fill in the righ
EE364 FALL 2014: H OMEWORK #5
Due date: Nov. 20, 2014 at the start of lecture (12:30pm)
P ROBLEM 1: A 150 mlong ship is being loaded with cargo boxes in two separate compartments on front and
the back. The front compartment can hold 20 cargo boxes and th
EE364 FALL 2014: Q UIZ #6
Name:
Student ID:
Instructions:
(1) Quiz is graded out of 10 with 15 possible points. 5 points can count as extra credit.
(2) Please show your work. This will help maximize any partial credit. Putting an answer without
showing wo
EE364 FALL 2014: T UTORIAL #2
Monday, Sept. 15, 2014
P RACTICE PROBLEMS
P ROBLEM 1 (Sample space and integration limits): Consider the probability distribution function
f (x) =
0
13
12
x
x<0
0x1
1<x
1 2
2
0
(a) If the sample space S is dened to be S = R
EE364 FALL 2014: T UTORIAL #12
Monday, Nov. 24, 2014
P ROBLEM 1: Suppose that the lifetime of Badger brand light bulbs is modeled by an exponential distribution with (unknown) parameter . We test 5 bulbs and nd they have lifetimes of 2, 3, 1, 3, and 4 yea
EE 364 Fall 2014
HW #1
Due date: Thursday, Sep. 11, 2014, beginning of class
1. List the elements of each of the following sample spaces:
(a) the set of integers between 1 and 100 that are divisible by 6 and 8.
(b) the set S = cfw_x  x2 3x 40 = 0.
(c) th
EE364 FALL 2014: H OMEWORK #2
Due date: September 25th at the start of lecture (12:30pm)
P ROBLEM 1: Consider discrete random variables X, Y with joint probability distribution
f (x, y) x = 1 x = 2 x = 3
y = 0 1/93 4/93 9/93
y = 1 4/93 9/93 16/93
y = 2 9/
Introduction to Probability and Statistics for Electrical Engineering
EE 364

Spring 2016
Lecture 4: Probability3
1. Additive Rules: We now step to the next level of complexityS and multiple
events.
a. Fact 1: Multiple events are combined using logical connectors.
b. Fact 2: We associate the word OR with the inclusive OR, i.e., with the
union
Introduction to Probability and Statistics for Electrical Engineering
EE 364

Spring 2017
LING275 Final Review
Reading
1. In English, the correspondences between letters and sounds is totally arbitrary.
a. True correspondence between visual word and its
b. False meaning
2. Refer to the figure above where C1, C2, and C3 are labels for the mapp
Introduction to Probability and Statistics for Electrical Engineering
EE 364

Spring 2017
52 views
note
Lecture #7  31 Jan 2017
0. Weekly quiz
Theorem on total probability,
.
1. The weakness in Bayesian reasoning: P[H], the prior probability
Consider thescenario where you are scanning the skies for signs of extraterrestrial (E.T.) intelligenc
Introduction to Probability and Statistics for Electrical Engineering
EE 364

Spring 2017
108 views
note
Lecture #1  10 Jan 2017
Distributed Syllabus
HW #1  due 17 Jan
1. Models
 A modelis an abstraction of reality with simplied behaviors and cause and eect. But it is only as good as the modeller (YOU!) and the assumptions that underlie the
Introduction to Probability and Statistics for Electrical Engineering
EE 364

Spring 2017
102 views
note
Lecture #2  12 Jan 2017
0. Review
Four problems with "frequentist" probability
Basic strucutre of probability theory
outcomes
sample spaces
events
probability as size
Set Theory
1.Sets and elements
Defn: A set is a collection of events.
Ex
Introduction to Probability and Statistics for Electrical Engineering
EE 364

Spring 2017
85 views
note
Lecture #6  26 Jan 2017
0. Review
Conditional probability
Independence
1. Theorem of total probability
Defn: The sets
partition the sample space
Thrm (Theorem of total probability): Suppose
if
are disjoint and
form a partition for
.
. Then
Introduction to Probability and Statistics for Electrical Engineering
EE 364

Spring 2017
EE 364 (68 unread)
4/3/17, 8)56 PM
96 views
note
Lecture #21  28 March 2017
0. Review
Markov inequality: If X is a nonnegative random variable then P [X
Chebyshev inequality (1): P [X 
(alternative form): P [X 
a]
k]
a]
E[X]
a .
2
.
a2
1
.
k2
Introduction to Probability and Statistics for Electrical Engineering
EE 364

Spring 2017
92 views
note
Lecture #23  04 April 2017
0. Review
Defn: The entropy of a random variable
is:
.
is expressed in units of "bits".
gives the amount of uncertainty resolved by observing .
1. Partial information
Ex: Suppose
equiprobable from the set:
.
Then
Introduction to Probability and Statistics for Electrical Engineering
EE 364

Spring 2017
12 views
note
Lecture #24  06 April 2017
0. Weekly quiz
Entropy and optimalbinary decision tree.
1. Review
Multiple random variables and events in higher dimensions
2. JointPMF
Defn: Suppose
is a pair of discrete randomvariables. The joint probability ma
Introduction to Probability and Statistics for Electrical Engineering
EE 364

Spring 2017
28 views
note
Lecture #26  13 April 2017
0. Weekly Quiz
Integrating joint pdf. Marginal pdf.
1. Conditional probability of multiple random variables
1.1 Discrete random variables
Defn: If
and
are discrete random variables then the conditional pmf of
give
Introduction to Probability and Statistics for Electrical Engineering
EE 364

Spring 2017
26 views
note
Lecture #25  11 April 2017
0. Review
Suppose
Joint pmf:
is a pair of random variables.
.
.
Marginal pmf:
Joint cdf:
.
Joint pdf:
if it exists.
and
.
and
.
.
Marginal pdf:
1. Joint pdf example
Ex:
. Find
(gure, omitted)
integrating pdf over