Solved by verified expert:hello, i will upload a practice problems with the answers provided and i will upload a practice exam with answers provided what i want is to solve every question with giving me the way to solve it so i can study from it please be quick.
practice_problems_for_test_1__2_.docx
practice_test_1_problems__1_.docx
answers_for_practice_problems_test_1__2_.docx
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Practice Problems for Test 1
Question 1: If the sample space, S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and the events in this sample
space are: A = {0, 2, 4, 6, 8}, B = {1, 3, 5, 7, 9}, C = {2, 3, 4, 5} and D = {1, 6, 7}
Then list the elements of the sets corresponding to the following events:
(a)
(b)
(c)
(d)
C∩D
(A ∩ B)´U C
C ∩ D´
C´∩ D´
Question 2: From past experience, a stockbroker believes that under present economic
conditions a customer will invest in tax-free bonds with a probability of 0.6, will invest in mutual
funds with a probability of 0.3, and will invest in both tax-free bonds and mutual funds with a
probability of 0.15. At his time find the probability that a customer will invest
(a) in either tax-free bonds or mutual funds?
(b) in neither tax-free bonds or mutual funds?
Draw a vend diagram.
Question 3: A regional telephone company operates three identical relay stations at different
locations. During a one-year period, the number of malfunctions reported by each station and
the causes are shown below.
Station
A
B
C
Problems with electricity supplied
2
1
1
Computer malfunction
4
3
2
Malfunctioning electrical equipment 5
4
2
Caused by other human errors
7
7
5
Suppose that a malfunction was reported and it was found to be caused by other human errors.
(a) What is the probability that it came from station C?
Question 4: A manufacturing firm employs three analytical plans for the design and development
of a particular product. For cost reasons, all three are used at varying times. In fact, plans 1, 2,
and 3 are used for 30%, 20%, and 50% of the products, respectively.
The defect rate is different for the three procedures as follows:
P (D|P1) = 0.01, P (D|P2) = 0.03, P (D|P3) = 0.02,
Where P (D|Pj) is the probability of a defective product, given plan j. (a) If a random product was
observed and found to be defective, which plan was most likely used and thus responsible
(explain)?
Question 5: An important factor in solid missile fuel is the particle size distribution. Significant
problems occur if the particle sizes are too large. From production data in the past, it has been
determined that the particle size (in micrometers) distribution is characterized by
3x −4 ,
x > 1,
f (x) = {
0, elsewhere.
(a) Verify that this is a valid density function.
(b) What is the probability that a random particle from the manufactured fuel exceeds 4
micrometers?
Question 6: How many ways can a student pick 3 of 5 English courses and 2 of 4 Philosophy Courses?
Question 7: Let X denote the reaction time, in seconds, to a certain stimulus and Y denote the
temperature (◦F) at which a certain reaction starts to take place. Suppose that two random variables X
and Y have the joint density
ì
ï 4xy,
f (x, y) = í
ï
0,
î
0 < x <1, 0 < y <1,
elsewhere.
(a) What is P(X<0.25); (b) What is the P(X<0.5 and Y<0.3)
Question 8: There is a club of 90 sports fans. 45 of the 90 go to football games, 40 of the 90 go to
basketball games. 30 go to both football and basketball. What is the probability if you picked one fan
that they went to football and not to basketball. Draw a Venn Diagram.
Question 9:
X
f(x,y)
0
1
2
h(y)
y| 0
3/28
9/28
3/28
15/28
y| 1
6/28
6/28
y|
g(x)
(a)
(b)
(c)
(d)
2
12/28
1/28
10/28
What is P(X<1 | y<2)?
What is P(Y=1| x<2)?
What is g(2)?
What is h(1)?
1/28
15/28
3/28
1
Question 10: Suppose that a family is leaving on a summer vacation in their camper and that M is the
event that they will experience mechanical problems, T is the event that they will receive a ticket for
committing a traffic violation, and V is the event that they will arrive at a campsite with no vacancies.
List the numbers of the regions that represent the following events:
(a) The family will experience no mechanical problems and will not receive a ticket for a traffic violation
but will arrive at a campsite with no vacancies.
(b) The family will experience both mechanical problems and trouble in locating a campsite with a
vacancy but will not receive a ticket for a traffic violation.
(c) The family will either have mechanical trouble or arrive at a campsite with no vacancies but will not
receive a ticket for a traffic violation.
(d) The family will not arrive at a campsite with no vacancies.
(e) What is missing from the Venn Diagram?
Question 11: (a) How many distinct permutations can be made from the letters of the word COLUMNS?
(b) How many of these permutations start with the letter M?
Question 12: A box contains 500 envelopes, of which 75 contain $100 in cash, 150 contain $25, and 275
contain $10. An envelope may be purchased for $25. (a) What is the sample space for the different
amounts of money? (b) Assign probabilities to the sample points and then find the probability that the
first envelope purchased contains less than $100.
Question 13: It is common in many industrial areas to use a filling machine to fill boxes full of product.
These machines are not perfect, and indeed they may A, fill to specification, B, underfill, and C, overfill.
Generally, the practice of underfilling is that which one hopes to avoid. Let P(B)=0 .001 while P(A)=0
.990. (a) Give P(C). (b) What is the probability that the machine does not underfill? (c) What is the
probability that the machine either overfills or underfills?
Question 14: Find the probability of randomly selecting 4 good quarts of milk in succession from a
cooler containing 20 quarts of which 5 have spoiled, show two different ways to calculate.
Question 15: Indicate whether the random variable is continuous or discrete.
(a)The number of eggs laid by a specific hen per month. (b) The weight of grain produced per acre.
(c)The length of time for an egg to hatch. (d) The number of farms in Nebraska.
PRACTICE TEST
1. Your manufacturing plants have the resources to support running 4 of 7 operations in a facility in Morgantown
and 3 of 6 operations in a facility in Pittsburgh. How many ways can the operations be scheduled?
2. There is a production operation that has three machines. Machine A makes 25% of the production, Machine B
makes 35% of the production and Machine C makes 40% of the production. Defect rates are: Machine A – 4%,
Machine B – 3% and Machine C – 3%.
a. If you bought one of these products, what is the probability it is defective?
b. Given you have a defect product what is the likelihood it came from Machine B?
c. If you were the plant engineer, which machine would you focus on improving first? Why?
3. What is the mean and median of each of the following sample data sets? What is standard deviation for a?
a. 7, 1,3,9,8
b. 2,3,1,4,6,9
4. Given the following Table 1:
f(x,y)
Y
1
2
3
g(x)
1
0.10
0.05
0.20
X
2
0.20
0.05
0.10
a. What is the value of h(3)?
b. What is the P(X < 2)?
c. What is the P(Y<2|X<3)?
?
d. What is the value of
3
0.05
0.10
0.15
h(y)
?
?
5. Given that f(x,y) = x + y for 0 < x <1 and 0 < y < 1
What is the P(0
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