Download probability of an event - hedge fund analysis

Last Update
6th April 2011
SESSION 27 & 28
Probability Theory
Lecturer:
University:
Domain:
Florian Boehlandt
University of Stellenbosch Business School
http://www.hedge-fundanalysis.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:
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-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:
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
0.29
Not top-10 Student
0.06
0.54
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
Distinction
No Distinction
Top-10 Student
0.11
0.29
Not 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:
Event B1
Event B2
Total
Event A1
Event A2
Total
Formally:
1
Marginal Probability - Example
From the previous example:
Distinction
No Distinction
Total
Top-10 Student
0.11
0.29
0.40
Not top-10
Student
0.06
0.54
0.60
Total
0.17
0.83
1.00
e.g. out of all students, 17% received a distinction. 60%
of all students do not belong to the Top-10 students.
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
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 top10 student(A2)?
Formula
Result
Conditional Probability - Example
From the previous example we wish to
determine the following:
Condition
Probability
required
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)?
Formula
Result
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
Student is a top-10
student (A1).
A student received
a distinction (B1).
Student not a top10 student(A2)?
A student received
a distinction (B1).
Formula
Result
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:
A1
A2
B1
0.20
0.15
B2
0.60
0.05
Hint: You require all marginal probabilities (4) and conditional
probabilities (8).
Exercise 1b
Are the events are independent given the following joint
probabilities?
A1
A2
B1
0.20
0.60
B2
0.05
0.15
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!
Exercise 2
A department store records mode of payment and money spent.
The joint probabilities are:
Cash
Credit Card
Debit Card
Under ZAR 50
0.05
0.05
0.04
50 – 200 ZAR
0.03
0.21
0.18
Over ZAR 200
0.09
0.23
0.14
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?
Exercise 3
Below you find the classifications of accounts within a firm:
Event A
Overdue
Not overdue
New
0.08
0.13
Old
0.50
0.29
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.