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CHAPTER OUTLINES:
3.1 Introduction
3.2 Sample Spaces And Probability Concepts
3.3 Marginal and Conditional Probabilities
3.4 Events & Probability Rules
3.4.1 Mutually Exclusive Events
3.4.2 Independent and Dependent Events
3.4.3 Complementary Events
OBJECTIVES:
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After completing this chapter, students you should be able to;
1. Determine sample spaces and find the probability of an
event, using classical probability or empirical probability.
2. Find the probability of compound events, using the
addition rules.
3. Find the probability of compound events, using the
multiplication rules.
4. Find the conditional probability of an event.
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The principles of probability help bridge the worlds of
descriptive statistics and inferential statistics.
Probability can be defined as the chance of an event
occurring or to be specific the numeric value representing
the chance, likelihood, or possibility a particular event
will occur.
Situations that involve probability:
1. Observing or playing a game of chance such as card
games and slot machines
2. Insurance
3. Investments
4. Weather Forecasting etc.
It is the basis of inferential statistics such as predictions
and testing the hypotheses
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Some basic concepts of probability:
1. A PROBABILITY EXPERIMENT
- A chance process that leads to well-defined results called outcomes.
2. AN OUTCOME
- The result of a single trial of a probability experiment.
3. A SAMPLE SPACE
- The set of all possible outcomes of a probability experiment.
- Some sample spaces for various probability experiments are shown below.
EXPERIMENT SAMPLE SPACES
Toss one coin Head, Tail
Roll a die 1, 2, 3, 4, 5, 6
Answer a true/false questions
True, False
Toss two coins Head-Head, Head-Tail, Tail-Tail, Tail-Head
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Example 1
Find the sample space for rolling two dice.
Die1Die 2
1 2 3 4 5 6
1 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
Example 2
Find the sample space for the gender of the children if
a family has three children. Use B for boy and G for
girl.
There are two genders, male and female and each child could be either gender. Hence, there are eight possibilities.
4. A TREE DIAGRAM
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B
G
B
B
G
G
B
G
B
G
B
G
B
G
1st child
2nd child
3rd child
Outcomes
- Another way to determine all possible outcomes (sample space) of a probability experiment.
- It is a device consisting of line segments emanating from a starting point and also from the outcome point.
Example 3
Use a tree diagram to find the sample space for the
gender of three children in a family.
Example 4
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1
2
3
You are at a carnival. One of the carnival games asks
you to pick a door and then pick a curtain behind the
door. There are 3 doors and 4 curtains behind each
door. Use a tree diagram to find the sample spaces for
all the possible choices.
5. VENN DIAGRAM
Curtain OutcomesDoor
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- developed by John Venn and are used in set theory and symbolic logic.
- have been adapted to probability theory.
- a picture (a closed geometric shape such as a rectangle, a square, or a circle) that depicts all the possible outcomes for an experiment.
- the symbol represents the union of two events and P(A B) corresponds to A OR B.
- the symbol represents the intersection of two events and P(A B) corresponds to A AND B.
6. AN EVENT
Venn Diagram representing two events; A and B
Venn Diagram representing three events; A, B and C
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- Consists of a set of outcomes of a probability experiment.
- An event can be :
a) Simple event – an event with one outcomee.g: If a die is rolled and a 6 shows since it is a
result of single trial
b) Compound event – an even with more than one outcome.e.g : The event of getting an odd number when
a die is rolled since it consists of three outcomes or three simple events.
Probabilities can be expressed as fractions, decimals or percentage (where appropriate).
There are four basic probability rules:
1. The probability of any event E is a number between and including 0 and 1.
0≤P( E )≤1
2. If an event E cannot occur, its probability is 0 (impossible event).
3. If an event is certain, then the probability of E is 1 (certain event).
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4. The sum of the probabilities of all the outcomes in the sample space is 1.
Three basic interpretations of probability that are used to solve a variety of problems in business, engineering and other fields:
1. CLASSICAL PROBABILITY
- Uses sample spaces to determine the probability an event will happen.
- Assumes that all outcomes in the sample space are equally likely to occur which means that all the events have the same probability of occurring.
- The probability of any event E is:
Number of outcomes in E
Total number of outcomes in the sample space
Or denoted as, P( E )= n(E )
n (S)
- E.g. : When a single die is rolled, each outcome has the same probability of occurring. Since there are
six outcomes, each outcome has a probability of 16 .
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2. EMPIRICAL PROBABILITY
- Relies on actual experience to determine the likelihood of outcomes.
- Is based on observation.
- Given a frequency distribution, the probability of an event being in a given class is:
Frequency for the class Total frequencies in the distribution
Or denoted as,
P( E )= fn
Example 5
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Hospital records indicate that maternity patients stayed in the hospital for the number of days shown in the following distribution:
Number of days stayed Frequency
3 154 325 566 197 5
Total 127
Find these probabilities,
a) A patient stayed exactly 5 days
b)A patient stayed less than 6 days
c) A patient stayed at most 4 days
3. SUBJECTIVE PROBABILITY
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- Uses a probability value based on an educated guess or estimate, employing opinions and inexact information.
- This guess is based on the person’s experience and evaluation of a solution.
- E.g: A physician might say that, on the basis of her diagnosis, there is a 30% chance the patient will need an operation.
In the previous topics we described how to produce a sample space and assign probabilities to the simple events in the sample space. In this topic, we discuss how to calculate probability of more complicated events from the probability of related events.
Field of Events
1. Intersection event Let A and B be two events defined in a sample
space.
The intersection of events A and B is the event that occurs when both A and B occur.
It is denoted by either A B or AB. Example 6
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A B
A = event that a family owns a DVD player
B = event that a family owns a digital camera
2. Union event
Let A and B be two events defined in a sample space.
The union of events A and B is the event that occurs when either A or B or both occur.
It is denoted as A B.
Example 7
A = event that a family owns a DVD player
B = event that a family owns a digital camera
Example 8
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A senior citizens centre has 300 members. Of them, 140 are male, 210 take at least one medicine on a permanent basis and 95 are male and take at least one medicine on a permanent basis. Draw a Venn diagram to describe,
a) the intersection of the events “male” and “take at least one medicine on a permanent basis”.
b) the union of the events “male” and “take at least one medicine on a permanent basis”.
c) the intersection of the events “female” and “take at least one medicine on a permanent basis”.
d) the union of the events “female” and “take at least one medicine on a permanent basis”.
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3. Independent event
Two events A and B are independent events if the fact that A occurs does not affect the probability of B occurring.
Example 9
Rolling a die and getting a 6, and then rolling a second die and getting a 3.
4. Dependent event
When the outcome or occurrence of the first event affects the outcome or occurrence of the second event in such a way that the probability is changed, the events are said to be dependent events.
Some examples of dependent events:
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a) Drawing a card from a deck, not replacing it, and then drawing a second card.
b) Selecting a ball from an urn, not replacing it, and then selecting a second ball.
c) Having high grades and getting a scholarship.
d) Parking in a no-parking zone and getting a parking ticket.
NOTE: THE EXAMPLES OF JOINT, MARGINAL AND CONDITIONAL PROBABILITIES WILL BE BASED ON THE FOLLOWING CONTIGENCY TABLE
Table 1: Gives the two-way classification of all employees of a company by gender and college degree
Category College graduate, G
Not a college
graduate,
Total
Male, M 7 20 27
Female, F 4 9 13
Total 11 29 40
Joint Probability
The probability of the intersection of events.
Written by either P(A B) or P(AB).
Example 10 Refer Table 1 (Page 17)
If one of those employees is selected at random for membership on the employee management committee,
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there are 4 joint probabilities that can be defined. That is,
a) the probability that this employee is a male and a college graduate
b) the probability that this employee is a female and a college graduate
c) the probability that this employee is a male and not a college graduate
d) the probability that this employee is a female and not a college graduate
Marginal Probability
The probability of a single event without consideration of any event.
Also called as simple probability.
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Named so as they calculated in the margins of the table (divide the corresponding totals for the row or column by the grand total).
Example 11 Refer Table 1 (Page 17)
If one of those employees is selected at random for membership on the employee management committee, find the probabilities for each of the followings:
a) the chosen employee is a male
b) the chosen employee is a female
c) the chosen employee a college graduate
d) the chosen employee is not a college graduate
Conditional Probability
- Often used to gauge the relationship between two events.
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- Conditional probability is the probability that an event will occur given that another event has already occurred.
- Written as:P(event will occur | event has already occur)
- The probability of event A given event B is
- The probability of event B given event A is
Example 12 Refer Table 1 (Page 17)
If one of those employees is selected at random for membership on the employee management committee, find the probabilities for each of the followings:
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a) the chosen employee is a male given that he is graduated from college
b) the chosen employee is not a college graduate given that this employee is female
Example 13
A person owns a collection of 30 CDs, of which 5 are country music.
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a) 2 CDs are selected at random and with replacement. Find the probability that the second CD is country music given that the first CD is country music.
b) This time the selection made is without replacement. Find the probability that the second CD is country music given that the first CD is country music.
3.4.1 Mutually Exclusive Events & Non-Mutually Exclusive Events
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Two events are mutually exclusive if they cannot occur at the same time (they have no outcomes in common).
The probability of two or more events can be determined by the addition rules.
There are two addition rules to determine either the two events are mutually exclusive or not mutually exclusive.
Addition Rule 1
When two events A and B are mutually exclusive, the probability that A or B will occur is
P(A or B) = P(A) + P(B) or P(A and B) = 0
Addition Rule 2
P(A) P(B)
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P(B)
A B
When two events A and B are not mutually exclusive, then P(A or B)= P(A) + P(B) – P(A and B)
Example 14
Consider the following events when rolling a die:A = an even number is obtained = 2,4,6B = an odd number is obtained = 1,3,5
Are events A and B are mutually exclusive ?
Solution:
Example 15
P(A)
P(A and B)
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Determine which events are mutually exclusive and which are not when a single die is rolled.
a) Getting a 3 and getting an odd number.
b) Getting a number greater than 4 and getting a number less than 4.
c) Getting an odd number and getting a number less than 4.
Example 16There are 8 nurses and 5 physicians in a hospital unit; 7 nurses and 3 physicians are females. If a staff person is selected, find the probability that the subject is a nurse or a male.
Solution:Staff Female, F Male, M Total
Nurses, N 7 1 8Physicians, PY 3 2 5
Total 10 3 13
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Example 17
At a convention there are 7 mathematics instructors, 5
computer sciences instructors, 3 statistics instructors,
and 4 science instructors. If an instructor is selected,
find the probability of getting a science instructor or a
math instructor.
Solution:
P(science instructor or math instructor)
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Example 18
A grocery store employs cashiers, stock clerks and deli personnel. The distribution of employees according to marital status is shown here.
Marital Status Cashiers Clerks Deli Personnel
Married 8 12 3
Not Married 5 15 2
If an employee is selected at random, find these probabilities:
a. the employee is a stock clerk or married
b. the employee is not married
c. the employee is a cashier or is unmarried
3.4.2 Independent vs Dependent Events
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For two independent events, A and B, the occurrence of event A does not change the probability of B occurring.
The probability of independent events can be determined as:
P( A | B ) = P(A) Or
P( B | A ) = P(B)
Or
Multiplication Rule 1When two events are independent, the probability of both occurring
P(A B) = P(A) P(B)
Example 19
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A box contains 3 red balls, 2 blue balls, and 5 white balls. A ball is selected and its colour noted. Then it is replaced. A second ball is selected and its colour noted. Find the probability of each of these:
a. selecting two blue balls.
b. selecting 1 blue ball and then 1 white ball.
c. selecting 1 red ball and then 1 blue ball.
Example 20
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A survey found that 68% of book buyers are 40 years or older. If two book buyers are selected at random, find the probability that both are 40 years or older.
On the other hand, two events, A and B are dependent when the occurrence of the event A changes the probability of the occurrence of event B.
When two events are dependent, another multiplication rule can be used to find the probability.
Multiplication Rule 2When two events are dependent, the probability of both occurring
P (A B) = P(A) P( B | A )
Example 21
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PG
PG
1st tigress
2nd tigress 3rd tigress Outcomes
PGPG
PG
PG
PG
PGPG
PG
PG
PG
PG
38
37
57
36
36
26 4
6
PG
In a scientific study there are 8 tigresses, 5 of which are
pregnant. If 3 are selected at random without
replacement, find the probability that:
a) all tigresses are pregnant.
P (PGPGPG) =
b) two tigresses are pregnant.
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Let A be an event of two tigresses are pregnant
P(A) = P(PGPGPG ) + P(PGPGPG)
+ P(PGPGPG)
=
3.4.3 Complementary Events
The set of outcomes in the sample space that is not
included in the outcomes of event E.
Denoted as E (read “E bar”)
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Example 22
Find the complement of each event.
a. Rolling a die and getting a 4
b. Selecting a letter of the alphabet and getting a vowel
c. Selecting a day of the week and getting a weekday
The outcomes of an event and the outcomes of the complement make up the entire sample space.
The rule of complementary events can be stated algebraically in three ways:
P( E )=1−P (E ) Or
P( E )=1−P (E ) Or
P( E )+ P( E)=1
The concept can be represented pictorially by the following Venn Diagram.
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P(E)
P(S)=1
P(E)
)(EP
Example 23
In a group of 2000 taxpayers, 400 have been audited by the IRS at least once. If one taxpayer is randomly selected from this group, what are the probability of that taxpayer has never been audited by the IRS?
Solution:
Let, A = the selected taxpayer has been audited by the IRS at least once
A = the selected taxpayer has never been audited by the IRS
The multiplication rules can be used with the complementary event rule to simplify solving probability problems involving “at least”.
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Example 24
At a local university 54.3% of incoming first year students have computers. If three students are selected at random, find the probability at least one has the computer
Solution:
Let, C = at least one student has a computer
C = none of the students has a computer
P(has computer) =
So, P(has no computers) =
By using the complementary event rule,
P(C )=1−P(C ) =
Example 25
In a department store there are 120 customers, 90 of whom will buy at least one item. If 4 customers are selected at random, one by one, find the probability that
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at least one of the customers will but at least one item. Would you consider this event likely to occur? Explain.
Solution: Let
C = at least one customer will buy at least one item
C = none of the customers will buy at least one item
P(will buy at least one item) =
So, P(won’t buy any items) =
By using the complementary event rule,
P(C )=1−P(C )
= 1−( 1
4×1
4×1
4×1
4 ) = 1 –
1256 =
255256 = 0.9961
Yes, this event is most likely to occur (certain event) since the probability almost 1
NOTE: THE FOLLOWING EXAMPLES ARE BASED ON THE OVERALL UNDERSTANDING OF THE ENTIRE PROBABILITY CONCEPTS
Example 26
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A random sample of 400 college students was asked if college athletes should be paid. The following table gives a two-way classification of the responses.
Should be paid, PAID
Should not be
paid, PAIDTotal
Student athlete, SA 90 10 100
Student non-athlete, SNA
210 90 300
Total 300 100 400
a) If one student is randomly selected from these 400 students, find the probability that this student
i. Is in favour of paying college athletes
ii. Favours paying college athletes given that the student selected is a non-athlete
iii. Is an athlete and favours paying student athletes
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iv. Is a non-athlete or is against paying student athletes
b) Are the events “student athlete” and “should be paid” independent? Are they mutually exclusive? Explain why or why not.
P(SAPAID) = 9/40 = 0.2250 and
P(SA) P(PAID) =
100 ×400
=
Since, P(SAPAID) P(SA) P(PAID), those two events are not independent (dependent).
And since P(SAPAID) 0, those two events are not mutually exclusive
Example 27
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A screening test for a certain disease is prone to giving false positives of false negatives. If a patient being tested has the disease, the probability that the test indicates a false negative is 0.13. If the patient does not have the disease, the probability that the test indicates a false positive is 0.10. Assume that 3% of the patients being tested actually have the disease. Suppose that one patient is chosen at random and tested. Find the probability that;
Let D = the patient has the diseaseD = the patient does not have the diseasePO = the patient tests positiveNE = the patient tests negative
a. This patient has the disease and tests positive
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b. This patient does not have the disease and tests positive
c. This patient tests positive
d. This patient does not have the disease and tests negative
e. This patient has the disease given that he/she tests positive
EXERCISES
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1. For each of the following, indicate whether the type of probability involved is an example of classical probability, empirical probability or subjective probability:
i) the next toss of a fair coin will land on heads.
ii) italy will win soccer’s World Cup the next time the competition is held.
iii) the sum of the faces of two dice will be 7.
iv) the train taking a commuter to work will be more than 10 minutes late.
2. A test contains two multiple-choice questions. If a student makes a random guess to answer each question, how many outcomes are possible? Draw a tree diagram for this experiment. (Hint: Consider two outcomes for each question – either the answer is correct or it is wrong).
3. Refer to question 1. List all the outcomes included in each of the following events and mention which are simple and which are compound events.i) Both answers are correct.ii) At most one answer is wrong.iii) The first answer is correct and the second is wrong.iv) Exactly one answer is wrong.
4. State whether the following events are independent or dependent.i) Getting a raise in salary and purchasing a new car.
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ii) Having a large shoe size and having a high IQ.
iii) A father being left-handed and a daughter being left-handed.
iv) Eating an excessive amount of ice cream and smoking an excessive amount of cigarettes.
5. 88% of American children are covered by some type of health insurance. If four children are selected at random, what is the probability that none are covered?
6. A box of nine golf gloves contains two left-handed gloves and seven right-handed gloves.i) If two gloves are randomly selected from the box
without replacement, what is the probability that both gloves selected will be right-handed?
ii) If three gloves are randomly selected from the box without replacement, what is the probability that all three will be left-handed?
iii) If three gloves are randomly selected from the box without replacement, what is the probability that at least one glove will be right-handed?
7. A financial analyst estimates that the probability that the economy will experience a recession in the next 12 months is 25%. She also believes that if the economy encounters
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recession, the probability that her mutual fund will increase in value is 20%. If there is no recession, the probability that the mutual fund will increase in value is 75%. Find the probability that the mutual fund’s value will increase.
8. A car rental agency currently has 44 cars available. 18 of which have a GPS navigation system. One of the 44 cars is selected at random, find the probability that this car,
i) has a GPS navigation system.
ii) does not have a GPS navigation system.
Now, two cars are selected at random from these 44 cars. Find the probability that at least one of these cars have GPS navigation system.
9. A recent study of 300 patients found that of 100 alcoholic patients, 87 had elevated cholesterol levels, and 200 non-alcoholic patients, 43 had elevated cholesterol levels.
i) If a patient is selected at random, find the probability that the patient is the following, an alcoholic with elevated cholesterol level. a non-alcoholic. a non-alcoholic with non-elevated cholesterol level.
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ii) Are the events “alcoholic” and “non-elevated cholesterol levels” independent? Are they mutually exclusive? Explain why or why not.
10. The probability that a randomly selected student from college is female is 0.55 and that a student works more than 10 hours per week is 0.62. If these two events are independent, find the probability that a randomly selected student is ai) male and works for more than 10 hours per week.
ii) female or works for more than 10 hours per week.
11. A housing survey studied how City Sun homeowners get to work. Suppose that the survey consisted of a sample of 1,000 homeowners and 1,000 renters.
Drives to Work Homeowner Renter
Yes 824 681
No 176 319
i) If a respondent is selected at random, what if the probability that he or she
drives to work?
drives to work and is a homeowner?
does not drive to work or is a renter?
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ii) Given that the respondent drives to work, what then is the probability that he or she is a homeowner?
iii) Given that the respondent drives to work, what then is the probability that he or she is a renter?
iv) Are the two events, driving to work and the respondent is a homeowner, independent?
v) Purchased more products and changed brands?
vi) Given that a consumer changed the brands they purchased, what then is the probability that the consumer purchased fewer products than before?
12. Due to the devaluation which occurred in country PQR, the consumers of that country were buying fewer products than before the devaluation. Based on a study conducted, the results were reported as the following:
BrandsPurchased
Number of Products Purchased
Fewer Same MoreSame 10 14 24Changed 262 82 8
What is the probability that a consumer selected at random:
i) purchased fewer products than before?
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ii) purchased the same number or same brands?
iv) purchased more products and changed brands?
iv) given that a consumer changed the brands they purchased, what then is the probability that the consumer purchased fewer products than before?
13. A soft-drink bottling company maintains records concerning the number of unacceptable bottles of soft drink from the filling and capping machines. Based on past data, the probability that a bottle came from machine I and was non-conforming is 0.01 and the probability that a bottle came from machine II and was non-confirming is 0.0025. If a filled bottle of soft drink is selected at random, what is the probability that
i) it is a non-confirming bottle?
ii) it was filled on machine I and is a conforming bottle?
iii) it was filled on machine II or is a conforming bottle?
iv) suppose you know that the bottle was produced on machine I, what is the probability that it is non-conforming?
14. Each year, ratings are compiled concerning the performance of new cars during the first 90 days of use.
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Based on a study, the probability that the new car needs a warranty repair is 0.04, the probability that the car manufactured by Country ABC is 0.60, and the probability that the new car needs a warranty repair and was manufactured by Country ABC is 0.025.
i) What is the probability that the car needs a warranty repair given that Country ABC manufactured it?
ii) What is the probability that the car needs a warranty repair given that Country ABC did not manufacture it?
iii) Are need for a warranty repair and country manufacturing the car statistically independent?
15. CASTWAY is a direct selling company which has 350 authorized sale agents from all over the country. It is known that 168 of them are male. 40% of male sale agents has permanent job while half of female sale agents do not have permanent job. i) Draw a tree diagram to illustrate the above events.
ii) What is the probability that a randomly selected sale agent, has permanent job? is a male given that he does not have permanent
job?
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