SESSION 27 & 28

Last Update6th April 2011

Probability Theory

Lecturer: Florian BoehlandtUniversity: University of Stellenbosch Business SchoolDomain: http://www.hedge-fund-analysis.net/pages/ve

ga.php

Learning Objectives

All measures for grouped data: 1. Assigning probabilities to events2. Joint, marginal, and conditional probabilities3. Probability rules and trees

Terminology

A random experiment is an action or process that leads to one of several possible outcomes.For example:

Experiment Outcome

Flip a coin Heads and tails

Record marks on stats test Number between 0 and 100

Record student evaluation Poor, fair, good, and very good

Assembly time of a computer Number with 0 as lower limit and no predefined upper limit

Political election Party 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(Oi) of any outcome must lie between 0 and 1. That is:

Sum-of-probabilitiesThe 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:

Experiment Probability outcome

Coin toss ½ = 50%

Tossing of a die ⅙ = 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

Distinction No Distinction

Top-10 Student 0.11 0.29Not top-10 Student 0.06 0.54

The following notation represent the events:A1 = Student is in the top-10 of the classA2 = Student is not in the top-10 of the classB1 = Student gets distinction on stats testB2 = Student does not get distinction on stats test

Joint Probability - Example

Distinction No Distinction

Top-10 Student 0.11 0.29Not top-10 Student 0.06 0.54

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 B1 Event B2 Total

Event A1

Event A2

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.

Distinction No Distinction Total

Top-10 Student 0.11 0.29 0.40Not top-10 Student 0.06 0.54 0.60

Total 0.17 0.83 1.00

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:

Condition Probability required

Formula Result

A student received a distinction (B1).

What is the probability that the student is a top-10 student (A1)?

A student received a distinction (B1).

What is the probability that the student isn’t a top-10 student(A2)?

Conditional Probability - Example

From the previous example we wish to determine the following:

Condition Probability required

Formula Result

A student is in the top-10 (A1).

What is the probability that a student receives a distinction (B1)?

A student is in the top-10 (A1)..

What is the prob. that a student doesn’t receive distinction (B2)?

Conditional Probability - Exercise

Calculate the remaining conditional probabilities

and complete the table below. Use complementary probabilities when possible!

Condition Probability required

Formula Result

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 A Event B Formula Result

Student is a top-10 student (A1).

A student received a distinction (B1).

Student not a top-10 student(A2)?

A student received a distinction (B1).

Union of Events - Exercise

Calculate the remaining probabilities for the unions

Event A Event B Formula Result

Exercise 1a

Determine whether the events are independent from the following joint probabilities:

Hint: You require all marginal probabilities (4) and conditional probabilities (8).

A1 A2

B1 0.20 0.15B2 0.60 0.05

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!

A1 A2

B1 0.20 0.60B2 0.05 0.15

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?

Cash Credit Card Debit Card

Under ZAR 50 0.05 0.05 0.0450 – 200 ZAR 0.03 0.21 0.18Over ZAR 200 0.09 0.23 0.14

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 A Overdue Not overdue

New 0.08 0.13Old 0.50 0.29