statistics lecture notes dr. halil İbrahim cebecİ chapter 05 probability

Statistics Lecture Notes

Dr. Halil İbrahim CEBECİ

Chapter 05Probability

Key components of the statistical inference process is probability because it provides the link between sample and population.

Random Experiment:

An action or process that leads to one of several possible outcomes E.g. Flip a coin (Heads and Tails), Record student

evaluations of a course (poor, fair, good, very good, excellent)

Assigning Probability to Events

Statistics Lecture Notes – Chapter 05

Sample Space:

A sample space of a random experiment is a list of all possible outcomes of the experiment. The outcomes must be exhaustive and Mutually exclusive

All the possible outcomes must be included (exhaustive) No two outcomes can occur at the same time (Mutually

exclusive)



Requirements of probebilities:

Given a , the probabilities assigned to the outcomes must satisfy two requirements:

1. The Probability of any outcome must lie between 0 and 1. That is ,

for each

2.The sum of the probabilities of all the outcomes in a sample space must be 1. That is,



Event:An event is a collection or set of one or more simple events is a sample space.E.g. Achieve grade of A ()

Probability of events:Sum of the probabilities of the simple events that constitute the event.

Ex5.1 – Probabilities of the courses grade are

Probability of the event, pass the course, is


Approaches to Assigning Probabilities

One way to interpret probability is this:

If a random experiment is repeated an infinite number of times, the relative frequency for any given outcome is the probability of this outcome.

For example, the probability of heads in flip of a balanced coin is .5, determined using the classical approach. The probability is interpreted as being the long-term relative frequency of heads if the coin is flipped an infinite number of times.


Interpreting Probability

Joint probability is the probability that two events will occur simultaneously. (Intersection of Events A and B is the event that occurs when both A and B occur.)

Ex5.2 – Suppose that a potential investor examined the relationship between how well the mutual fund performs and where the fun manager earned his or her MBA. Analyze the probabilities given below and interpret the results.


Joint Probability (Intersection)

Mutual Fund Outperforms Market

Mutual Fund does not Outperforms Market

Top 20 MBA Programs 0.11 0.29

Not Top 20 MBA Programs 0.06 0.54

A5.2 – Evens notaion presented below.

Joint probabilities are;


Joint Probability (Intersection)

Marginal probability is the probability of the occurrence of the single event


Marginal Probability

Mutual Fund Outperforms Market

Mutual Fund does not Outperforms Market Totals

Top 20 MBA Programs

Not Top 20 MBA Programs

Totals

Marginal Probablilites

Conditional probability is used to determine how two events are related; that is, we can determine the probability of one event given the occurrence of another related event.

Conditional probabilities are written as and read as “the probability of event A given event B” and is calculated as:


Conditional Probability



Male Female TotalsAccounting 170 110 280

Finance 120 100 220

Marketing 160 70 230

Management 150 120 270

Totals 600 400 1000

Ex5.3 - The Dean of the School of Business at Owens University collected the following information about undergraduate students in her college:

Given that the student is a female, what is the probability that she is an accounting major?



A5.3 – Random experiment given below

One of the objectives of calculating conditional probability is to determine whether two events are related.

In particular, we would like to know whether they are independent, that is, if the probability of one event is not affected by the occurrence of the other event.

Two events A and B are said to be independent


Independence

Ex5.4 – Refer to table of Ex5.2, calculate probability of funds outperforms the market when the manager graduation probability is given.

The marginal probability that a manager graduated from a top-20 MBA program is

Since the two probabilities are not equal, we conclude that the two events are dependent.


Independence

Another event that is the combination of other events is the union

Union of Events and is the event that occurs when either or or both occur.

Ex5.5 – Refer to table of Ex5.2, Determine the probability that a ramdomly selected fund outperforms the market or the manager graduated from a top-20 MBA programs.

Shortcut:


Union

Complemet Rule:

The complement of an event A is the event that occurs when A does not occur.

The complement rule gives us the probability of an event NOT occurring. That is:

For example, in the simple roll of a die, the probability of the number “1” being rolled is . The probability that some number other than “1” will be rolled is .


Probability Rules and Trees

Multiplication Rule:

The multiplication rule is used to calculate the joint probability of two events. It is based on the formula for conditional probability defined earlier:

If we multiply both sides of the equation by we have:

Likewise,

If A and B are independent events, then



𝑃 ( 𝐴|𝐵 )= 𝑃 (𝐴𝑎𝑛𝑑𝐵)𝑃 (𝐵)

Addition Rule:

The Probability that event A, or event B, or both occur is

Why do we subtract the joint probability P(A and B) from the sum of the probabilities of A and B?



When two events are mutually exclusive

Probability Trees:

An effective and simpler method of applying rules is the probability tree, wherein the events in an experiment are represented by lines.

Ex5.6 – Student who graduate from law school must still pass a bar exam. First time test takers passes the exam with the ratio of 72%. Candidates who fail the first exam may take it again. Second time test takers passes with ratio of 88%. Find the probability that a randomly selected student becomes a lawyer.



Fail 0.12

Pass 0.88

Fail 0.28

Pass 0.72

First ExamSecond Exam

Pass (0.72) 0.72

Fail and Pass (0.28*0.88) 0.2464

Fail (0.28*0.12) 0.0336

Joint Probability

Bayes’ Theorem is a method for revising a probability given additional information.


Bayes’ Theorem

)/()()/()(

)/()()|(

2211

111 ABPAPABPAP

ABPAPBAP

Ex5.7 - Duff Cola Company recently received several complaints that their bottles are under-filled. A complaint was received today but the production manager is unable to identify which of the two Springfield plants (A or B) filled this bottle. What is the probability that the under-filled bottle came from plant A?


Bayes’ Theorem

% of total production % of underfilled bottleA 55 3B 45 4

A5.7 - Duff Cola Company recently received several


Bayes’ Theorem

4783.)04(.45.)03(.55.

)03(.55.

)/()()/()(

)/()()/(

BUPBPAUPAP

AUPAPUAP

The likelihood the bottle was filled in Plant A is .4783.

Q5.1 - A manufacturing plant conducted a survey to determine its employees’ reactions toward a proposed change in working hours. A breakdown of the responses is shown in the following table:

Suppose an employee is chosen at random, with the relevant events being defined as follows:

A: The employee works in production.B: The employee agrees with the proposed change.

Express each of the following events in words, and find the probabilities a) b) c) d)


Exercises

Work Area Agree Disagree

Production 17 23

Office 8 2

Reaction

Q5.2 - A manufacturing plant conducted a survey to determine its employees’ reactions toward a proposed change in working hours. A breakdown of the responses is shown in the following table:

One employee is selected at random, and two events are defined as follows:

A: The employee is male.B: The employee has worked for the company for two years or more.

find the following probabilities

a) b) c) d)


Exercises

Men Women

Less than 2 years 28 26

2 years or more 82 64

Q5.3 - An accounting firm has advertised the availability of its report describing recent changes to the federal income tax act. The first 200 callers requesting a copy of the report are classified in the following table according to the medium by which the caller became aware of the report and the caller’s primary interest.

One caller is selected at random, and two events are:A: The caller is primarily interested in corporate tax.B: The caller became aware of the report through the newspaper.

Express each of the following probabilities in words, and find its numerical value:

a) b) c) d)


Exercises

Primary Interest Radio Newspaper Word of Mouth

Personel Tax 34 20 26

Coorporate Tax 36 70 14

Q5.4 - A firm’s employees were surveyed to determine their feelings toward a new dental plan and a new life insurance plan. The results showed that 81% favored the insurance plan, while only 35% favored the dental plan. Of those who favored the insurance plan, 30% also favored the dental plan.

a. What percentage of the employees favored both plans?b. What percentage of the employees favored at least one

of the plans?


Exercises

Q5.4 - Consider two events, A and B, for which , , and

a. Find b. Are A and B independent events?c. Are A and B mutually exclusive events?

Q5.5 - An electrical contractor has observed that 90% of his accounts are paid within 30 days. Of those that are not paid within 30 days, 40% remain unpaid after 60 days. If one account is selected at random, what is the probability that it is paid within 60 days?

Q5.6 - A mechanic has removed six spark plugs from an engine and finds two to be defective. If two spark plugs are selected at random from among these six, what is the probability that exactly one of them is defective?


Exercises

statistics lecture notes dr. halil İbrahim cebecİ chapter 05 probability

Documents

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