April 4, 2009 - Class Notes

A. Basic Concepts of Probability
a) Defining probability
b) Probability experiment
c) Types of probability
d) Fundamental concepts related to probability
o The law of large numbers
o Range of probabilities rule
o Subjective probability
o Complement of an event E
B. Conditional Probability and the Multiplication Rule
a) Defining conditional probability
b) Defining multiplication rule
C. The Addition Rule
a) Mutually exclusive events

In units 2 and 3, you learned about collecting and describing data. In this unit, you will
strengthen your foundation in descriptive statistics by learning the concepts of probability.
In this unit, you will learn about various types of probability and the method of calculating
probability using various rules.


A. Basic Concepts of Probability

Probability refers to the likelihood of the occurrence of uncertain events.
A probability experiment is a trial through which specific results or outcomes are obtained.
An event consists of one or more outcomes and is a subset of the sample space. A simple event
has a single outcome, such as rolling a dice and obtaining 4.
Classical or theoretical probability refers to the type of probability when each outcome in a
sample space is equally likely to occur.
The classical probability of occurrence of an event E is given by:
Empirical or statistical probability is based on observations obtained from probability
experiments.
The empirical probability that an event E will occur is given by:
n
f
P(E) =
where,
f is the frequency of the event E occurring.
n is the total frequency of the experiment. n is sometimes denoted as Σf.
The law of large numbers
According to the law of large numbers, if an experiment is performed repeatedly, the empirical
Number of outcomes in an event
( )=
Total number of outcomes in a sample space
P E

probability of an event will be close to its theoretical or actual probability.
Range of probabilities rule
According to this rule, the probability of an event E is always between 0 and 1. Mathematically, it
is expressed as: 0 ≤ P(E) ≤ 1
Subjective probability
It describes an individual's personal judgment about the likelihood of the occurrence of an event.
It is based on estimates, intuition, and educated guess.
Complement of an event E
It refers to the set of all outcomes in a sample space that are not included in an event E. It is
denoted as E’—pronounced E prime. The probability of the complement of an event E is
calculated as follows:
P(E’) = 1 – P(E)
B. Conditional Probability and the Multiplication Rule
Conditional probability is the probability of an event B occurring, given that another event A
has already occurred. It is denoted by P(B|A).
Independent events do not affect the probability of occurrence of another event. For example,
getting a 2 after rolling a dice and getting a 2 on the next roll are independent events.
When two events A and B are independent, then:
P(B|A) = P(B) and P(A|B) = P(A)
In other words, if two events are independent, then
P(Aand B)=P(A)iP(B)
Dependent events are not independent.
The multiplication rule is used to determine the probability of the occurrence of two events A
and B in sequence.
The formula for multiplication rule is represented as follows:
P(A and B) = P(A) · P(B|A). However, if the events are independent, this formula reduces to
P(Aand B) =P(A)iP(B) .
C. Addition Rule
Two events are mutually exclusive if they have no outcomes in common. In other words, when
events are mutually exclusive, they cannot occur at the same time.
The addition rule is used to find the probability of occurrence of event A or B. Mathematically,
the addition rule is represented as follows:
P(A or B) = P(A) + P(B) – P(A and B), where P(A and B) is the probability of events A and B
occurring at the same time. If the events A and B are mutually exclusive P(A and B) = 0, and the
formula reduces to P(A or B) = P(A) + P(B).