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Project
| Major | Subject | Additional Information | Due Date |
| Math Education | Lesson Plan Involving Probability |
Project
Lesson Plan Lesson Plan Rubric Permission Slip This is a permission slip that I found online. Comments? |
Friday, April 24 |
| Business | Please see the Instructor |
Research Paper Format
|
Friday, April 24 |
| Engineering | Please see the Instructor |
Research Paper Format
|
Friday, April 24 |
Homework
If a problem requires work, it must be shown to receive credit.
Please submit your problem set in the order listed below.
All assignments are due by 10:30 am unless stated otherwise.
If you submit on Blackboard (pdf format only), you will receive a 5%
bonus.
| Homework | Problems | Key | Due Date |
| 1 |
1.2 (This question is asking for one example of a specific outcome and event, not a general statement of what all possible answers
could be) (p.33),
#1: "Five chairs are placed in a row; two of them
are red and look identical, two of them are blue and look identical; the last one is green.
What is the sample space?", 1.5 (p.34), 1.8 (p.34),
#2: "Suppose that one 8-sided die is rolled and that you observe the number of dots
facing up. a) Obtain a sample space for this random experiment whose elements are integers. b) Determine as a
subset of the sample space each of the following events: A = die comes up even, B = die comes up at least 4,
C = die comes up at most 2, and D = die comes up 3. c) Determine as a subset of the sample space and describe
in words each of the events Ac, A ∩ B, B U C, and B\A. d) Describe in words each of the events: see part b) for
examples of what is wanted:
{5}, {1, 3, 5}, and {1, 2, 3, 4}. #3: "Show
that both of DeMorgan's Laws (Theorems 1.22, 1.23) are valid for two sets via Venn Diagrams. You will need to
have a separate diagram for each complement, union and intersection." #4:
"Show that both distributive laws are valid using Venn Diagrams. You will need to show
each union and intersection."
|
Key |
Jan. 23 |
| 2 |
#5:
"Use the frequentist interpretation
of probability to interpret each statement. a) The probability is 0.314 that the gestation period of a woman will
exceed 9 months. b) The probability is 2/3 that the favorite in a horse race finishes in the money (first, second,
or third place). c) The probability is 0.40 that a traffic fatality involves an intoxicated or alcohol-impaired
driver or person not in the car."
#6:
"This is a continuation of #2 (Homework 1) using the same sets
A, B, C and D defined in that problem. Determine which of the following collections of events are disjoint
and explain your answers: A and B, B and C, A, C, and D.", #7:
"(Be sure to indicate why each statement is true like in a geometric proof) Let S be the sample space
Use the axioms to verify the following for events A and B. For part b) you may use the result in a), for part c) you may
use the results in a) and b): a) P(B) = P(B ∩ A) + P(B ∩ Ac)
b) P(A U B) = P(A) + P(B ∩ Ac), c) Supposed that at least one of the events A and B must occur
-- that is, A U B = S. Show that the probability that both events occurs is P(A) + P(B) - 1."
#8:"The sample space for the rolling of one 8-sided die is
S = {1, 2, 3, 4, 5, 6, 7, 8}. The table provides five different potential probability
assignments to the possible outcomes. a) Which of the assignments #1 - #5 are legitimate probability assignments?
Explain. b) Determine the probability that the die comes up odd by using each of the legitimate probability assignments
in the table. c) If the die is balanced, which probability assignment should be used. Explain.",
2.2 (p.49), 2.14 (b is more difficult,p.52), 2.20 (p.52),
2.8 (assume a maximum wait time of 30 mins., p.51),
#9:"Assuming the scenario in problem 2.8,
what is the
probability that the wait time will be at least 10 mins?"
2.22 (p.53), #10: "For the following
situations, identify whether the type of probability is equally-likely, empirical or subjective. Please explain
your answer.
a) The percentage of male students in STAT 311 in Spring 2015 is 43.75%.
b) The percentage of male students in STAT 311 in Spring 2016 will be less than 50%.
c) A student puts 4 books in her backpack; 3 of them are math books and 1 is a statistics book. Assume that all of the books are approximately
the same size. If she reaches into her backpack without looking, the probability of her choosing the statistics book is 1/4.
d) In 2013, 65.9% of all high school graduates were enrolled in colleges or universities.
#11: "A person is selected at random from among the inhabitants of a state.
Which is more probable: that the person so chosen is a teacher, or the person so chosen is a female teacher?
Explain your answer." |
|
Key
|
Jan. 30 |
| 3 |
2.12 (p.51), 2.24 (p.53), 2.26 (p. 54),
3.8 (p. 64), #12:
"In this exercise, you will go over the concepts of independent and disjoint events.
Some of these were
done in class, but I want you to go over them again. a) If two events
are disjoint, determine their joint probability. b) If two events with positive probability are independent,
explain why their joint probability is not 0. The conclusion is that these two events can't be disjoint. c) Give
an example (which was not given in class or the book) of two events that are neither disjoint nor independent.
d) Give
an example (which was not given in class or the book) of two events that are disjoint, but not independent.
e) Give an
example (which was not given in class or the book) of two events that are independent, but not disjoint.
f) Give an example
(which was not given in class or the book) of two events that are both independent and disjoint."
3.4 (p.63), 3.10 (p.64), #13:
"Respond true
or false. If true, provide the proof, if false, provide a counter example (that was not used
in class). If events A and B are
independent and events B and C are independent, then events A and C are independent." #14:
"The game of craps is played by rolling two balanced dice. A first roll
of a sum of 7 or 11 wins; a first roll of a 2, 3, or 12 loses. To win with any other first sum, the sum
must be repeated before a sum of 7 is thrown. Suppose that a player rolls a sum of 4 on the first roll.
Determine the probability that the player wins." 4.2 (p.74), 4.4 (p. 75),
4.10 (p.76) |
|
Key
|
Feb. 6 |
| 4 |
5.2 (p. 86),
#15 |
Key |
Feb. 13 |
| 5 |
#21: "Derive the formula for the ordered partition, SLIDE Chapter 4: Slide 11."
21.1 (p.293),
21.10 (p.294), 21.12 (p.294), 21.22bc (p.296),
21.26 (p.296), 21.30 (p.297),
#22: "In tossing 5 6-sided fair dice, what is the probability of at
least one 2? a) If the dice are distinguishable? b) If the dice are indistinguishable? #23:
"You roll 6 dice where order is important, the first two are 40sided, the next two are 60sided and the last two are 10-sided.
a) How many different combinations are there? b) What is the total number of combinations if no number can occur twice on
the size size die? (The last part of this question is #16 on the combinatorics 2 worksheet.)" #24:
"The answers to all of the parts of Worksheet Combinatorics 2". #25: Poker hands (22.1)
a) Straight Flush,
b) Two Pair, c) Three of a kind", #26: Yahtzee (22.2) a) Four of a Kind, b) Small Straight |
|
Key |
Feb. 27 |
| 6 |
6.2 (p.98), 6.8 (p.99), 6.14 (p.99),
6.16 (p.99), Plots in this assignment
may be done by hand or via computer software. If they are done by hand, be sure that the scales are appropriate.
7.4 (p. 113), 7.6 (p.114), 7.8 (p.114),
7.10 (p.115), #27:
"Using the mass and
CDF determined in exercise 7.6, calculate the following probabilities using both the mass and CDF (this means
that you will be calculating each probability twice) a) the probability that you find at most 2 watches, b) the
probability that you find less than 2 watches, c) the probability that you find between 3 and 7 watches exclusive,
d) the probability that you find between 3 and 7 watches inclusive."
8.10 (p.134), #28:
"In problem 8.10, a) Find the marginal masses for both X and Y.
b) Find the marginal CDFs for both X and Y.
Determine each of the following probabilities c) P(X = Y), d) P(X < Y), e) P(X > Y).
#29: "In problem 8.10, are X and Y independent random variables? Explain your answer."
#30:
"An automobile insurance policy reimburses a family's accident losses up to a
maximum of two accidents per year. For a (high-risk) two person family, let X and Y denote the number of
accidents in a single year for the two people. Assume the joint mass of X and Y is
pX,Y(x,y) = (x + 2y)/15 where x ∈ {0,1}
and y ∈ {0,1,2} and 0 else. a) Determine the conditional joint PMF of Y given X = 0. b) Determine
the conditional joint PMF of Y given X = 1. c) Given that X = 1, during a particular year, find the probability
that all of the family's automobile accidents will be reimbursed. c) Are X and Y independent random variables?
Explain your answer."
|
|
Key |
March 6 |
| 7 |
9.2 (p.141), 9.8 (p.142), 9.14
(p.143, Note: the mass was generated in 8.10 (p. 134)), #31: "9.19 (p.144), except use
4-sided dice instead of 6-sided dice."
#32: "A random sample of size n is taken from a very large
lot of items in which 100p1% have exactly one defect and 100p2% have two or more defects, where
0 < p1 + p2 < 1. An item with exactly one defect costs $1 to repair, whereas an item with two or
more defects costs $3 to repair. Determine the expected cost of repairing the defective items in the sample."
#33: "A box contains 10 disks of radii 1, 2, ..., 10, respectively.
What is the expected value of the area of a disk selected at random from the box?" 10.2 (p.151),
10.10 (p.152),
#34: "Prove that if P(X = c) = 1 for some c, then Var(X) = 0. Hint:
Use the definition of variance and let Y = (X - μX)2.", #35:
"Prove Corollary 11.20 (without using Theorem 11.19) a)
Var(bX) = b2Var(X) b) Var(X + a) = Var(X) Hint: Use the definition of variance and rules for
expected values." #36: "For a standardized random variable X*,
a) Calculate the expected value of X*. b) Calculate Var(X*)."
11.2 (p.168), #37: "11.4bc (p.168) except use a 4-sided die. Part a) was computed
in 9.14", 11.14 (p.170),
#38: This is a continuation of 11.14.
"a) What is the variance of the price of a drink and an appetizer when
all drinks are $2
(plus tax)? b) What is the variance of the price of 5 independent people who order appetizers (with drinks)?" c) If W is the random variable in
part a). Let Z = 5W. What is the variance of Z?"
|
|
Key |
March 13 |
| SPRING BREAK - Mar. 16 - Mar. 20 | |||
| 8 |
13.4 (p.191), 13.8 (p.192),
14.2 (p.203), 14.4 (p.204), 14.6 (p.204),
17.2abcde (all but part f, p.245), 17.4 (p.246), 17.16 (p.249),
#39: "Using the story in 17.16, a) Why is the approximate probability
appropriate?
b) Calculate the theoretical probability and compare to your result in 17.16 b). Hint: If your calculator
can not perform the calculations, then use
Wolfram Alpha."
|
|
Key |
March 27 |
| 9 |
15.2 (part g): make a plot similar to Example 15.1 e), however, you only need to plot every 5th point.
in part h). you may just place points (instead of lines) for every 5th point. Remember that h) is a CDF. I will post the complete graphs
in the key. p.220),
15.10 (p.222), #40:
"Using the information in 15.10, a) What is the probability that X is between 4 and 8 inclusive?
b) What is the probability that X is between 4 and 8 exclusive? c) How many pieces would he have to pull out
until he has a 92% or greater chance of getting a blue moon ?", 15.12 (p.223), 15.18 (p.224),
16.2 (in parts g) and h), just plot every 4th point. I will post the complete graphs in the key. p.231-232),
16.6 (p.233), 16.8 (p.233), #41:
"Given the situation in 16.8, what is the probability that he will find exactly 10 women who fit the
glass slipper if he checks 75 women?",
18.2 (p.259, Note: parts g and i are duplicates, you only need to do the part once),
18.6 (p.260),
#42: "Using the situation in 18.6, a) Solve part d),
remember Wolfram Alpha, b)
Why is the approximation valid? c) Calculate the variances using the original distribution and the approximation.
d) Compare the probabilities, expected values and variances between the calculations using the hypergeometric
distribution and the approximation, would you consider the approximation appropriate. Why or why not?
(Please use a table for the comparison.)", 18.8 (p.261),
18.14 (p.262),
18.18 (p.253),
19.4 (Expected value: required, variance: bonus, p.267),
19.2 (p.267), #43: "State which distribution is being described in each
situation in the document and why. Note: A correct answer can be
'none', but you still need to explain why." |
|
Key |
April 3 |
| 10 |
23.2 (p.324), 23.4 (p.324),
23.8abcdgh (p.325),
23.14 (p.326), #44: "Suppose a random variable
X has a density given by k(2x - 1) over the values 1 < x < 3 and 0 else. Find k so that this function is a
valid density.", #45: "Suppose a random variable X has the following density, (7x/4)
from 0 ≤ x ≤ 1, (1/8) from 7 ≤ x ≤ 8 and 0 else. Find the CDF."
24.5 (p.339), #46:: For the density in 24.5, find the joint CDF., 24.6 (p.339),
#47: "Verify that the density in problem 24.6 is legitimate.", 24.8 (pp. 339-340),
#48: "In 24.8, are X and Y independent? Answer this questions using remark 25.5 and definition 25.1",
#49: "Determine the joint density when FX,Y(x,y) = x2y/2 + xy2/2 when 0 < x < 1 and 0 < y < 1",
25.2 (Answer this question using remark 25.5 and definition 25.1, p.349),
25.5 (Answer this question using remark 25.5 and definition 25.1, p.350), 25.20 (p.353)
|
|
Key |
MONDAY April 20 at 5 pm |
| 11 |
26.3 (p.361), #50 In problem 26.3, given y = 2, what is the conditional probability
that 0 ≤ x ≤ 1? 26.16 (p.363), 28.14 (p.390),
28.16 (p.391), 28.19 (p.391), 28.22 (p.391)
#51: "Derive the density for a continuous uniform random variable on the interval [a,b] give them
the density function is a constant, that is fX(x) = c (a constant)." 30.2 (p.415-416),
30.12 (Hint: What is the distribution? p.418),
#52: "Derive the CDF for an exponential random
variable.",
31.2 (p.434), 31.6 (p.435), 31.12 (p.436),
34.3 (p.482), 34.6 (p.483),
34.10 (p.483), 34.16 (p.484),
34.22 (p.485)
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|
Key |
WEDNESDAY April 29 |
Worksheet Keys:
Worksheet Joint Continuous KeyWorksheet conditional, expectations, review V
Worksheet Normal
Worksheet Normal Approximation to the Binomial
Worksheet Review Named Distributions Key
