SESSION 27 & 28 Last Update 6 th April 2011 Probability Theory

Lecturer:Florian Boehlandt University:University of Stellenbosch Business School Domain: analysis.net/pages/vega.php

Learning Objectives All measures for grouped data: 1.Assigning probabilities to events 2.Joint, marginal, and conditional probabilities 3.Probability rules and trees

Terminology A random experiment is an action or process that leads to one of several possible outcomes. For example: ExperimentOutcome Flip a coinHeads and tails Record marks on stats testNumber between 0 and 100 Record student evaluationPoor, fair, good, and very good Assembly time of a computerNumber with 0 as lower limit and no predefined upper limit Political electionParty A, Party B, …

Assigning Probabilities 1.Produce a list of outcomes that is exhaustive (all possible outcomes must be accounted for) and mutually exclusive (no two outcomes may occur at the same time). The sample space of a random experiment is then the list of all possible outcomes. 2.Assign probabilities to the outcomes imposing the sum-of-probabilities and non-negativity constraints.

Requirements of Probabilities Non-negativity: The probability P(O i ) of any outcome must lie between 0 and 1. That is: Sum-of-probabilities The sum of all k probabilities for all outcomes in the sample space must be 1. That is:

Assigning Probabilities (cont.) The classical approach is used to determine probabilities associated with games of chance. For example: ExperimentProbability outcome Coin toss½ = 50% Tossing of a die⅙ = 16.67% Probability of winning the lottery

Assigning Probabilities (cont.) The relative frequency approach defines probability as the long-run relative frequency with which outcomes occur. The probabilities represent estimates from the sample and improve with larger sample sizes. When it is not reasonable to use the classical approach and there is not history of outcomes (or too short a history), the subjective approach is employed (‘judgment call’).

More Terminology An event is a collection or set of one or more simple events in the sample space. In the stats grade example, an event may be defined as achieving a distinction grade. In set notation, that is:

More Terminology The probability of an event if the sum of probabilities of the simple events that constitute the event. For example, the probability that tossing a die will yield four or below: Assuming a fair die, the probability of said event is:

Joint Probability The intersection of events A and B is the event that occurs when both a and B occur. The probability of the intersection is called joint probability. Notation:

Joint Probability – Example COPY DistinctionNo Distinction Top-10 Student Not top-10 Student The following notation represent the events: A1 = Student is in the top-10 of the class A2 = Student is not in the top-10 of the class B1 = Student gets distinction on stats test B2 = Student does not get distinction on stats test

Joint Probability - Example DistinctionNo Distinction Top-10 Student Not top-10 Student The joint probabilities are then: Note that the sum of the joint probabilities = 1.

Marginal Probability - COPY Marginal probabilities are calculated by adding across the rows and down the columns: Formally: Event B 1 Event B 2 Total Event A 1 Event A 2 Total 1

Marginal Probability - Example From the previous example: e.g. out of all students, 17% received a distinction. 60% of all students do not belong to the Top-10 students. DistinctionNo DistinctionTotal Top-10 Student Not top-10 Student Total

Conditional Probability The conditional probability expresses the probability of an event given the occurrence of another event. The probability of event A given event B is: Conversely, the probability of event B given A is:

Conditional Probability - Example From the previous example we wish to determine the following - COPY: ConditionProbability required FormulaResult A student received a distinction (B 1 ). What is the probability that the student is a top-10 student (A 1 )? A student received a distinction (B 1 ). What is the probability that the student isn’t a top- 10 student(A 2 )?

Conditional Probability - Example From the previous example we wish to determine the following: ConditionProbability required FormulaResult A student is in the top-10 (A 1 ). What is the probability that a student receives a distinction (B 1 )? A student is in the top-10 (A 1 ).. What is the prob. that a student doesn’t receive distinction (B 2 )?

Conditional Probability - Exercise Calculate the remaining conditional probabilities and complete the table below. Use complementary probabilities when possible! ConditionProbability required FormulaResult

Independent Events Two events A and B are said to be independent if: Or: From the example: i.e. the event that a student is a top-10 student is not independent of the performance on the test.

Union of Events The union of events A and B is the event that occurs when either A or B or both occur: Formally, this may be calculated either using the joint probabilities: Or marginal probabilities and the joint probability:

Union of Events - Example From the previous example we wish to determine the following - COPY: Event AEvent BFormulaResult Student is a top-10 student (A 1 ). A student received a distinction (B 1 ). Student not a top- 10 student(A 2 )? A student received a distinction (B 1 ).

Union of Events - Exercise Calculate the remaining probabilities for the unions Event AEvent BFormulaResult

Exercise 1a Determine whether the events are independent from the following joint probabilities: Hint: You require all marginal probabilities (4) and conditional probabilities (8). A1A2 B B

Exercise 1b Are the events are independent given the following joint probabilities? Note that if then: Thus, in problems with only four combinations, if one combination is independent, all four will be independent. This rule applies to this type of problems only! A1A2 B B

Exercise 2 A department store records mode of payment and money spent. The joint probabilities are: a)What proportion of purchases was paid by debit card? b)What is the probability that a credit card purchase was over ZAR 200? c)Determine the proportion of purchases made by credit card or debit card? CashCredit CardDebit Card Under ZAR – 200 ZAR Over ZAR

Exercise 3 Below you find the classifications of accounts within a firm: One account is randomly selected: a)If the account is overdue, what is the probability that it is new? b)If the account is new, what is the chance that it is overdue? c)Is the age of the account related to whether it is overdue? Explain. Event AOverdueNot overdue New Old