MATH 431 Midterm: 2014 Math 431 - Anderson - Spring Exam 1
Math 431 - Fall 2013 February 19, 2014, 7:15PM - 8:45PM
First Midterm Exam
Name:
Please circle your section number
Section 1 Section 2 Section 3 Section 4
David Anderson Gregory Shinault David Anderson Scott Hottovy
(MWF 9:55) (TuTh 1) (MWF 1:20) (TuTh 9:30)
•There are seven problems on the exam, some of them have multiple parts.
•Read the problems carefully and budget your time wisely.
•If you are running out of time, please at least say *how* you would solve a given problem, preferably
using notation, even if you do not have the time to do so. This may earn you partial credit.
•No calculators or other electronic devices, please. Turn off your phone.
•You do not have to carry out complicated numerical computations, but you should simplify your answer
if it is possible with reasonable effort. In particular, geometric series must be evaluated.
•Please present your solutions in a clear manner. Justify your steps. A numerical answer without expla-
nation cannot get credit. Cross out the writing that you do not wish to be graded on.
•Use the empty pages for additional space for writing if necessary.
Problem Points
1/15
2/13
3/12
4/12
5/15
6/15
7/18
Total /100
1
1. (15 pts.) Suppose P(A) = 0.3 and P(B) = 0.6.
(a) (5 pts.) If Aand Bare disjoint, what is P(A∪B)?
(b) (5 pts.) If Aand Bare independent, what is P(A∪B)?
(c) (5 pts.) Now assume that P(A) = 0.4 and P(B) = 0.7. Making no further assumptions on Aand
B, prove that P(AB) satisfies 0.1≤P(AB)≤0.4.Full credit will only be given for clearly
written solutions!
2
2. (13 pts.) Suppose an urn contains three black chips, two red chips, and one green chip. Suppose the
chips are drawn one at a time until all are drawn. Let Abe the event that the first three chips drawn
are all of different colors.
(a) (3 pts.) Give an appropriate sample space, Ω, for this experiment.
(b) (10 pts.) Find P(A).
3
Document Summary
This may earn you partial credit: no calculators or other electronic devices, please. Turn o your phone: you do not have to carry out complicated numerical computations, but you should simplify your answer if it is possible with reasonable e ort. In particular, geometric series must be evaluated: please present your solutions in a clear manner. A numerical answer without expla- nation cannot get credit. Cross out the writing that you do not wish to be graded on: use the empty pages for additional space for writing if necessary. Suppose p (a) = 0. 3 and p (b) = 0. 6. (a) (5 pts. ) If a and b are disjoint, what is p (a b)? (b) (5 pts. ) If a and b are independent, what is p (a b)? (c) (5 pts. ) Now assume that p (a) = 0. 4 and p (b) = 0. 7. B, prove that p (ab) satis es 0. 1 p (ab) 0. 4.
