Wk2 DQ - Data Analysis & Business Intelligence

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A Survey of Probability Concepts

Chapter 5

5-1

This is the first chapter of inferential statistics. Statistical inference deals with conclusions about a population based on a sample taken from that population. First we learn the different definitions of statistics and then how to calculate probabilities. Later in the chapter, we learn how to calculate the total number of outcomes of an experiment.

Learning Objectives

LO5-1 Define the terms probability, experiment, event, and outcome

LO5-2 Assign probabilities using a classical, empirical, or subjective approach

LO5-3 Calculate probabilities using the rules of addition

LO5-4 Calculate probabilities using the rules of multiplication

LO5-5 Compute probabilities using a contingency table

LO5-6 Calculate probabilities using Bayes’ theorem

LO5-7 Determine the number of outcomes using principles of counting

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Probability

PROBABILITY A value between 0 and 1 inclusive that represents the likelihood a particular event happens.

5-3

Probability is a numerical value that describes the chance that something will happen. The closer the probability is to 0, the more improbable. The closer the probability is to 1, the more likely it will happen.

Probability (2 of 2)

EXPERIMENT A process that leads to the occurrence of one and only one of several possible results.

OUTCOME A particular result of an experiment.

EVENT A collection of one or more outcomes of an experiment.

5-4

Three key terms used in the study of probability are experiment, outcome, and event. An experiment is the observation of some activity or the act of taking some measurement, like tossing a coin or asking a question. In this example, the experiment is rolling a die. There are six possible outcomes and numerous possible events.

Classical Probability

The classical definition of probability applies when there are n equally likely outcomes to an experiment

MUTUALLY EXCLUSIVE The occurrence of one event means that none of the other events can occur at the same time.

COLLECTIVELY EXHAUSTIVE At least one of the events must occur when an experiment is conducted.

5-5

There are 3 definitions of probability. The classical approach to probability is often applied to games of chance like playing cards and rolling dice. It can also be applied to lotteries, because the total number of outcomes is known before the experiment. If the set of events are collectively exhaustive and mutually exclusive, the sum of their probabilities is 1.

Empirical Probability

The empirical definition occurs when the number of times an event happens is divided by the number of outcomes

EMPIRICAL PROBABILITY The probability of an event happening is the fraction of the time similar events happened in the past.

LAW OF LARGE NUMBERS Over a large number of trials, the empirical probability of an event will approach its true probability.

5-6

Empirical probability is also known as relative frequency. An example of a firm using empirical probability is when life insurance companies use past data to determine the acceptability of an applicant as well as the premium to be charged. Stephen Curry of the Golden State Warriors made 278 out of 302 free throw attempts during the 2017–18 NBA season. Based on the empirical approach to probability, the likelihood of him making his next free throw attempt is .92.

Subjective Probability

Examples of subjective probability are:

Estimating the likelihood the New England Patriots will be in the Super Bowl next year

Estimating the likelihood the U.S. budget deficit will be reduced by half in the next 10 years

SUBJECTIVE CONCEPT OF PROBABILTIY The likelihood (probability) of a particular event happening that is assigned by an individual based on whatever information is available.

5-7

If there is little or no data available, probability can be estimated subjectively. A subjective probability is based on whatever information is available; an individual evaluates the available information and then assigns a probability.

Summary of Approaches to Probability

5-8

Here is a summary of the different approaches to assigning probabilities.

Rules of Addition

The rules of addition refer to the probability that any two or more events can occur

The special rule of addition is used when the events are mutually exclusive

5-9

If the events A and B are mutually exclusive, the probability of one or the other event occurring is the sum of their probabilities. This can be extended for more than two events. Here is a Venn diagram illustrating that events A, B, and C do not overlap and are therefore mutually exclusive. See the textbook for more information on Venn diagrams.

Rules of Addition Example

A machine fills plastic bags with a mixture of beans, broccoli, and other vegetables. Most of the bags contain the correct weight, but because of the variation in the size of the beans and other vegetables, a package might be underweight or overweight. A check of 4,000 packages filled in the past month revealed:

What is the probability that a particular package will be either underweight or overweight?

P(A or C) = P(A) + P(C) = .025 + .075 = .10

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Note, the events A, B, and C are mutually exclusive and exhaustive.

Complement Rule

The complement rule is used to determine the probability of an event happening by subtracting the probability of an event not happening

You can also use the complement rule

P(A or C) = P(~B) = 1 − P(B) = 1 − .900 = .10

5-11

This rule is useful because sometimes it is easier to calculate the probability of an event happening by determining the probability of it not happening and subtracting the result from 1. The events need to be mutually exclusive and exhaustive.

General Rule of Addition

The general rule of addition is used when the events are not mutually exclusive

JOINT PROBABILITY A probability that measures the likelihood two or more events will happen concurrently.

5-12

The rules of addition are used to find the probability of two or more events occurring. Here the two events, A and B, overlap and illustrate the joint probability of A and B.

General Rule of Addition Example

A sample of 200 tourists in Florida shows 120 went to Disney, 100 went to Busch Gardens, and 60 visited both.

P(Disney) =120/200 = .60

P(Busch) =100/200 = .50

P(Disney and Busch) = 60/200 = .30

P(Disney or Busch) = P(Disney) + P(Busch) – P (Disney and Busch)

= .60 + .50 - .30 = .80

5-13

So that we do not double count, subtract the probability of those tourists that went to both attractions.

Special Rule of Multiplication

The rules of multiplication are applied when two or more events occur simultaneously

The special rule of multiplication refers to events that are independent

INDEPENDENCE The occurrence of one event has no effect on the probability of the occurrence of another event.

A survey by the American Automobile Association (AAA) revealed 60% of its members made airline reservations last year. Two members are selected at random. What is the probability both made airline reservations last year?

P(R1 and R2) = P(R1)P(R2) = (.60)(.60) = .36

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A joint probability is the likelihood that two or more events will happen at the same time. In the AAA example, the events are independent, because whether a member made an airline reservation or not has no effect on whether another member made an airline reservation. Therefore the special rule of multiplication can be used.

General Rule of Multiplication

The general rule of multiplication refers to events that are not independent

A conditional probability is the likelihood an event will happen, given that another event has already happened

The conditional probability is represented a P(B|A) and is read, the probability of B given A

CONDITIONAL PROBABILITY The probability of a particular event occurring, given that another event has occurred.

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When events are not independent, they are dependent. This is called a conditional probability because its value is conditional on what occurred with the first event. In other words, the probability of B is conditional on the occurrence and effect of event A.

General Rule of Multiplication Example

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So the likelihood of selecting two shirts and finding them both to be white is .55. This can be extended to more than two events.

Contingency Tables

CONTINGENCY TABLE A table used to classify sample observations according to two or more identifiable categories or classes.

One hundred fifty adults were asked if they were older than 50 years of age and the number of Facebook accounts they used. The following table summarizes the results.

5-17

A contingency table is a cross-tabulation that simultaneously summarizes two variables of interest and their relationship. The level of measurement can be nominal.

Tree Diagrams

A tree diagram is a visual that is helpful in organizing and calculating probabilities for problems with several stages

Each stage of the problem is represented by a branch of the tree

Label the branches with the probabilities

5-18

Table 5-1 summarizes data from a survey conducted by the National Association of Theater Managers. Five hundred randomly selected adults were asked their age and the number of times they saw a movie in a theater. The survey results are organized in a contingency table by age and by the number of movies attended per month. See the next slide for an illustration of how to use this data in a tree diagram to calculate probabilities.

Tree Diagram Example

5-19

This tree diagram summarizes all the probabilities based on the data in table 5-1. This information is useful for making decisions regarding discounts on tickets and concessions.

Bayes’ Theorem

Bayes’ Theorem is a method of revising a probability, given that additional information is obtained

For two mutually exclusive and collectively exhaustive events

PRIOR PROBABILITY The initial probability based on the present level of information.

POSTERIOR PROBABILITY A revised probability based on additional information.

5-20

A1 and A2 are mutually exclusive and exhaustive categories. See the next slide for an example.

Bayes’ Theorem Example

Suppose 5% of the population of Umen have a disease. A1 represents the part of the population that has the disease and A2 represents those who do not. Let B denote a test result that shows the disease is present.

P(A1) = .05 Individual has the disease

P(A2) = .95 Individual does not have the disease

P(B|A1) = .90 Test shows the individual has the disease and is correct

P(B|A2) = .15 Test incorrectly shows the individual has the disease

5-21

The probabilities of A1 and A2 are prior probabilities since their probabilities are assigned before any empirical data are obtained. P(A1) = .05 and let A2 represent the part that does not have the disease. P(A2) = 1-.05=.95 B denotes the event “test shows the disease is present.” Assume the probability for a positive test result for someone with the disease is .90 and the probability of a positive test result is .15 for an individual who does not have the disease. The probability P(A1|B) is called a posterior probability because it is a revised probability based on additional information. Using Bayes’ theorem, we find the probability of someone who tests positive and actually has the disease. Therefore, the probability that a person has the disease, given that he or she tested positive, is .24.

Bayes’ Theorem Example continued

5-22

Using Bayes’ theorem, we find the probability of someone who tests positive and actually has the disease. Therefore the probability that a person has the disease, given that he or she tested positive, is .24.

Multiplication Formula

The multiplication formula states that if there are n ways of doing one thing, and m ways of doing another thing, then there are m*n ways of doing both

This can be extended to more than two events. For three events m, n, and o:

Total number of arrangements = (m)(n)(o)

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There are three counting formulas that are useful in determining the number of outcomes in an experiment. This is the multiplication formula that is used to find the total number of arrangements for two or more groups.

Multiplication Formula Example

5-24

When the American Red Cross receives a blood donation, the blood is analyzed and classified by group and Rh factor. There are four blood groups: A, B, AB, and O. The Rh factor can be either positive or negative. How many different blood types are there?

Total possible arrangements = (m) (n) = (4) (2) = 8

The Permutation Formula

Another counting formula used to determine a total number of outcomes

PERMUTATION Any arrangement of r objects selected from a single group of n possible objects.

Label the parts A, B, and C ABC BAC CAB ACB BCA CBA

5-25

Use the permutation formula when the order of the objects is important and to find the number of r objects selected from a group of n objects. Remember, by definition, 0! = 1. Excel has a formula that will calculate permutations.

The Combination Formula

Another counting formula useful in determining the total number of outcomes

A combination is an arrangement where the order of the objects selected is not important

The Grand 16 movie theater uses teams of three employees to work the concession stand each evening. There are seven employees available to work. How many different teams can be scheduled?

5-26

The combination formula is used when the order of the objects is not important; it is used to count the number of r object combinations from a set of n objects. Logically, the number of combinations is always less than the number of permutations. In this example, the seven employees taken three at a time would create the possibility of 35 different teams.

Chapter 5 Practice Problems

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Question 3

5-28

A survey of 34 students at the Wall College of Business showed the following majors:

From the 34 students, suppose you randomly select a student.

What is the probability he or she is a management major?

Which concept of probability did you use to make this estimate?

LO5-2

Question 5

5-29

In each of the following cases, indicate whether classical, empirical, or subjective probability is used.

A baseball player gets a hit in 30 out of 100 times at bat. The probability is .3 that he gets a hit in his next at bat.

A seven-member committee of students is formed to study environmental issues. What is the likelihood that any one of the seven is randomly chosen as the spokesperson?

You purchase a ticket for the Lotto Canada lottery. Over five million tickets were sold. What is the likelihood you will win the $1 million jackpot?

The probability of an earthquake in northern California in the next 10 years above 5.0 on the Richter Scale is .80.

LO5-2

Question 13

5-30

A study of 200 advertising firms revealed their income after taxes:

What is the probability an advertising firm selected at random has under $1 million in income after taxes?

What is the probability an advertising firm selected at random has either an income between $1 million and $20 million, or an income of $20 million or more? What rule of probability was applied?

LO5-3

Question 21

5-31

The aquarium at Sea Critters Depot contains 140 fish. Eighty of these fish are green swordtails (44 female and 36 male) and 60 are orange swordtails (36 female and 24 males). A fish is randomly captured from the aquarium:

What is the probability the selected fish is a green swordtail?

What is the probability the selected fish is male?

What is the probability the selected fish is a male green swordtail?

What is the probability the selected fish is either a male or a green swordtail?

LO5-3

Question 25

5-32

A local bank reports that 80% of its customers maintain a checking account, 60% have a savings account, and 50% have both. If a customer is chosen at random:

What is the probability the customer has either a checking or a savings account?

What is the probability the customer does not have either a checking or a savings account?

LO5-4

Question 29

5-33

Each salesperson at Puchett, Sheets, and Hogan Insurance Agency is rated either below average, average, or above average with respect to sales ability. Each salesperson also is rated with respect to his or her potential for advancement—either fair, good, or excellent. These traits for the 500 salespeople were cross-classified into the following table.

What is this table called?

What is the probability a salesperson selected at random will have above average sales ability and excellent potential for advancement?

Construct a tree diagram showing all the probabilities, conditional probabilities, and joint probabilities.

LO5-5

Question 35

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The Ludlow Wildcats baseball team, a minor league team in the Cleveland Indians organization, plays 70% of their games at night and 30% during the day. The team wins 50% of their night games and 90% of their day games. According to today’s newspaper, they won yesterday. What is the probability the game was played at night?

LO5-6

Question 43

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An overnight express company must include five cities on its route. How many different routes are possible, assuming that it does not matter in which order the cities are included in the routing?

LO5-7