NAME
STUDENT #
York University
MATH 2030 3.00MW Elementary Probability
First Midterm Examination Solutions (Salisbury)
February 11, 2011
You have 50 minutes to complete this exam. There are 5 questions on 5 pages, worth
a total of 70 points. If you ru
MATH 2030 3.0MW (Elementary Probability)
Assignment 2 Solutions
January 2011 Salisbury
1.4 No. 4
(a) By independence, P (Ac B c ) = P (Ac )P (B c ) = (1 P (A)(1 P (B ) =
0.9 0.7 = 0.63
(b) A B = (Ac B c )c (ie having at least one occur means that it ISNT
York University
MATH 2030 3.0MW (Elementary Probability)
Assignment 3 Solutions
February 2011 Salisbury
1.5 No. 3
(a) We test a chip. Let B be the event that the chip is good, so B c is the event
that the chip is bad. Let A be the event that the chip pass
York University
MATH 2030 3.0MW (Elementary Probability)
Assignment 4 Solutions
February 2011 Salisbury
3.1 No. 9
X = 2 if the 2nd draw matches the rst, which has probability 1 . X = 3 if the
7
2nd draw diers from the rst, and the third matches the 1st or
York University
MATH 2030 3.0MW (Elementary Probability)
Assignment 4 Solutions
February 2011 Salisbury
3.1 No. 9
X = 2 if the 2nd draw matches the rst, which has probability 1 . X = 3 if the
7
2nd draw diers from the rst, and the third matches the 1st or
York University
MATH 2030 3.0MW (Elementary Probability)
Assignment 5 Solutions
February 2011 Salisbury
2.1 No. 4
By the denition of conditional probability, and the independence of the rolls,
P (2 sixes in rst 5 rolls | 3 sixes in 8 rolls)
P (2 sixes in
York University
MATH 2030 3.0MW (Elementary Probability)
Assignment 6 Solutions
March 2011 Salisbury
3.2 No. 5
(a) I dont think much of the argument. Write K for the number of times
the chosen number comes up and Ai for the event that the chosen number
1
York University
MATH 2030 3.0MW (Elementary Probability)
Assignment 7 Solutions
March 2011 Salisbury
3.3 No. 2
The distribution of Y is
y
0
1
2
3
P (Y = y ) 1/8 3/8 3/8 1/8
Recall that
E [g (Y )] = g (0)P (Y = 0) + g (1)P (Y = 1) + g (2)P (Y = 2) + g (3)P
York University
MATH 2030 3.0MW (Elementary Probability)
Assignment 8 Solutions
April 2011 Salisbury
2.4 No. 4
(a) Let X be the number of days with thirty or more 6s. Then X Bin(n, p)
where n = 365 and p = 0.00068; We approximate X by Y Poisson()
where =
York University
MATH 2030 3.0MW (Elementary Probability)
Practice problems Solutions
April 2011 Salisbury
3.1 No. 2
(a) The following gives P (X = x, Y = y ) for sampling with replacement.
1
1
2
3
4
y
2
3
4
1
16
1
16
1
16
1
16
1
16
1
16
1
16
1
16
1
16
1
1
York University
Faculty of Science and Engineering
MATH 2030 3.00MW Elementary Probability
Instructor: T. Salisbury
Second Midterm Examination Solutions
March 18, 2011
NAME:
SIGNATURE:
STUDENT NUMBER:
Instructions:
(1) You have 50 minutes to complete this
York University
MATH 2030 3.0MW (Elementary Probability)
Assignment 1 Solutions (corrected)
January 2011 Salisbury
1.1 No. 2
(a) We use an equally likely outcomes model with
= cfw_suppose, a, word, is, picked, at, random, from, this, sentence.
This has 1
MATH 2030 3.00MW Elementary Probability
Course Notes
Part V: Independence of Random Variables,
Law of Large Numbers,
Central Limit Theorem,
Poisson distribution
Geometric & Exponential distributions
Tom Salisbury [email protected]
York University
Winter 2010
York University MATH 2030 3.0AF (Elementary Probability) Midterm I - SOLUTIONS October 12, 2007 NAME: STUDENT NUMBER: You have 50 minutes to complete the examination. There are 5 questions on 4 pages. You may bring one letter-sized two-sided formula sheet
York University MATH 2030 3.0AF (Elementary Probability) Midterm Exam I SOLUTIONS October 17, 2008 NAME: STUDENT NUMBER: You have 50 minutes to complete the examination. There are 4 pages, containing 5 questions. You may bring one letter-sized two-sided f
York University MATH 2030 3.0AF (Elementary Probability) Midterm 1 Solutions October 4, 2006 Salisbury NAME: STUDENT NUMBER: You have 50 minutes to complete the examination. There are ve questions, on three pages. No other books or notes may be used. You
York University MATH 2030 3.0AF (Elementary Probability) Midterm 2 - Corrected Solutions November 7, 2007 NAME: STUDENT NUMBER: You have 50 minutes to complete the examination. There are 5 pages, containing 5 questions and a Normal table. You may bring on
York University MATH 2030 3.0AF (Elementary Probability) Midterm 2 October 30, 2006 Salisbury NAME: STUDENT NUMBER: You have 50 minutes to complete the examination. There are four questions, on three pages. No other books or notes may be used. You may use
York University MATH 2030 3.0AF (Elementary Probability) Class Quiz SOLUTIONS September 26, 2008 NAME: STUDENT NUMBER: You have 20 minutes to complete the quiz. There are 2 pages, containing 2 questions in total. You may bring one letter-sized two-sided f
MATH 2030 3.00MW Elementary Probability
Course Notes
Part I: Models and Counting
Tom Salisbury [email protected]
York University
Winter 2010
Introduction [Jan 5]
Probability: the mathematics used for Statistics
Are related but dierent.
Probability question: A
MATH 2030 3.00MW Elementary Probability
Course Notes
Part II: Independence and Conditional
Probabilities
Tom Salisbury [email protected]
York University
Winter 2010
Independence and Conditional Probabilities [Jan 19]
Events A and B are said to be independent
MATH 2030 3.00MW Elementary Probability
Course Notes
Part III: Random Variables
Tom Salisbury [email protected]
York University
Winter 2010
Discrete Distributions [Jan 31]
Recall that a r.v. has a discrete distribution if there are only
nitely or countably ma
MATH 2030 3.00MW Elementary Probability
Course Notes
Part IV: Binomial/Normal distributions
Mean and Variance
Tom Salisbury [email protected]
York University
Winter 2010
Binomial Distribution [Feb 14]
The course now swings towards studying specic distribution
MATH 2030 3.00MW Elementary Probability
Course Notes
Part V: Independence of Random Variables,
Law of Large Numbers,
Central Limit Theorem,
Poisson distribution
Geometric & Exponential distributions
Tom Salisbury [email protected]
York University
Winter 2010