Chapter 4:Probability

LO1 Describe what probability is and when one would use it.

LO2 Differentiate among three methods of assigning probabilities: the classical method, relative frequency of occurrence, and subjective probability.

LO3 Deconstruct the elements of probability by defining experiments, sample spaces, and events, classifying events as mutually exclusive, collectively exhaustive, complementary, or independent; and counting possibilities.

L04 Compare marginal, union, joint, and conditional probabilities by defining each one.

Learning Objectives

LO5 Calculate probabilities using the general law of addition, along with a joint probability table, the complement of a union, or the special law of addition if necessary.

LO6 Calculate joint probabilities of both independent and dependent events using the general and special laws of multiplication.

LO7 Calculate conditional probabilities with various forms of the law of conditional probability, and use them to determine if two events are independent.

LO8 Calculate conditional probabilities using Baye’s rule.

Learning Objectives

Probability

The theory of probability provides the statistical basis for estimating a parameter with a statistic.

There are three methods for assigning probabilities:

• The classical method (mathematical rules and laws)• Relative frequency of occurrence (based on

historical data: empirical )• Subjective probability (based on personal intuition

or reasoning)

Methods of Assigning Probabilities

• Number of outcomes leading to the event divided by the total number of outcomes possible

• Each outcome is equally likely• Determined a priori -- before performing the experiment• Applicable to games of chance• Objective -- everyone correctly using the method assigns an

identical probability

Classical Probability

• Based on historical data, not on rules or laws• Computed after performing the experiment• Number of times an event occurred divided by the number

of trials• Objective -- everyone correctly using the method assigns an

identical probability

Relative Frequency Probability

• Subjective probability comes from a person’s intuition or reasoning

• However different individuals may (correctly) assign different numeric probabilities to the same event

• Expresses an individual’s degree of belief• Useful for unique (single-trial) experiments

– New product introduction– Initial public offering of common stock– Site selection decisions– Sporting events

Subjective Probability

• Experiment• Event• Elementary Events• Sample Space• Unions and Intersections• Mutually Exclusive Events• Independent Events• Collectively Exhaustive Events• Complementary Events

Fundamental Concepts and Laws of Probability Theory

• Experiment: a process that produces outcomes– More than one possible outcome– Only one outcome per trial

• Trial: one repetition of the process• Elementary Event: cannot be decomposed or broken down

into other events• Event: an outcome of an experiment– May be an elementary event, or– May be an aggregate of elementary events– Usually represented by an uppercase letter, e.g., A, E1

Experiment

Experiment: randomly select, without replacement, two families from the residents of Tiny Town

• Elementary Event: the sample includes families A and C• Event: each family in the sample has children in the

household• Event: the sample families own a total of four automobiles

An Example Experiment

• A roster or listing of all elementary events for an experiment

• Methods for describing a sample space– roster or listing– tree diagram– set builder notation– Venn diagram

Sample Space

• Experiment: randomly select, without replacement, two families from the residents of Tiny Town

• Each ordered pair in the sample space is an elementary event, for example -- (D,C)

Sample Space: Roster Example

Sample Space: Tree Diagram for Random Sample of Two Families

• S = {(x,y) | x is the family selected on the first draw, and y is the family selected on the second draw}

• Concise description of large sample spaces

Sample Space: Set Notation for Random Sample of Two Families

• Useful for discussion of general principles and concepts

Sample Space

• The union of two sets contains an instance of each element of the two sets.

Union of Sets

X Y

• The intersection of two sets contains only those element common to the two sets.

Intersection of Sets

• Events with no common outcomes

• Occurrence of one event precludes the occurrence of the other event

Mutually Exclusive Events

• Occurrence of one event does not affect the occurrence or nonoccurrence of the other event

• The conditional probability of X given Y is equal to the marginal probability of X.

• The conditional probability of Y given X is equal to the marginal probability of Y.

Independent Events

• Contains all elementary events for an experiment

Collectively Exhaustive Events

• All elementary events not in the event ‘A’ are in its complementary event.

Complementary Events

• mn Rule

• Sampling from a Population with Replacement

• Combinations: Sampling from a Population without Replacement

Counting the Possibilities

• If an operation can be done in m ways and a second operation can be done in n ways, then there are mn ways for the two operations to occur in order.

• A cafeteria offers 5 salads, 4 meats, 8 vegetables, 3 breads, 4 desserts, and 3 drinks. A meal consists of one serving of each of the items. How many meals are available?

• (Ans: 548343 = 5,760 meals.)

mn Rule

• A tray contains 1,000 individual tax returns. If 3 returns are randomly selected with replacement from the tray, how many possible samples are there?

• (N)n = (1,000)3 = 1,000,000,000

Sampling from a Population with Replacement

• This counting method uses combinations

• Selecting n items from a population of N without replacement

Combinations: Sampling from a Population without Replacement

• For example, suppose a small law firm has 16 employees and three are to be selected randomly to represent the company at the annual meeting of the Bar Association.

• How many different combinations of lawyers could be sent to the meeting?

• Answer: NCn = 16C3 = 16!/(3!13!) = 560.

Combinations: Sampling from a Population without Replacement

• Marginal Probability

• Union Probability

• Joint Probability

• Conditional Probability

Four Types of Probability

Four Types of Probability

General Law of Addition

General Law of Addition -- Example

Office Design ProblemProbability Matrix

Office Design ProblemProbability Matrix

Office Design ProblemProbability Matrix

Venn Diagram of the X or Y but not Both Case

The Neither/Nor Region

The Neither/Nor Region

Special Law of Addition

Law of Multiplication Demonstration Problem 4.5

Law of Multiplication Demonstration Problem 4.5

• General Law

• Special Law

Special Law of Multiplication for Independent Events

• The conditional probability of X given Y is the joint probability of X and Y divided by the marginal probability of Y.

Law of Conditional Probability

Law of Conditional Probability

Office Design Problem

• If X and Y are independent events, the occurrence of Y does not affect the probability of X occurring.

• If X and Y are independent events, the occurrence of X does not affect the probability of Y occurring.

Independent Events

Independent EventsDemonstration Problem 4.10

Independent EventsDemonstration Problem 4.11

• An extension to the conditional law of probabilities• Enables revision of original probabilities with new information

Revision of Probabilities: Bayes’ Rule

Revision of Probabilities with Bayes' Rule: Over-the-Counter Problem

Revision of Probabilities with Bayes’ Rule: Over-the-Counter Problem

Revision of Probabilities with Bayes' Rule: Over-the-Counter Problem

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