Probability

Definitions
A probability is a measure of the likelihood
that an event in the future will happen. It
can only assume a value between 0 and
1.
 A value near zero means the event is
not likely to happen. A value near one
means it is likely.
 There are three ways of assigning
probability:
◦ classical,
◦ empirical, and
◦ subjective.

Basic Statements About
Probability
1. The probability, P, of any event or state of nature
occurring is greater than or equal to 0 and less than or
equal to 1.
That is: 0 P(event) 1
2. The sum of the simple probabilities for all possible
outcomes of an activity must equal 1.
3. Probability „p‟ of the happening of an event is also known
as probability of success & „q‟ the non-happening of the
event as the probability of failure.

4. If P(E) = 1, E is called a certain event &
if P(E) = 0, E is called an impossible event

Simple Definitions
 Trial & Event

◦ Example: - Consider an experiment which, though
repeated under essentially identical conditions, does
not give unique results but may result in any one of
the several possible outcomes.

◦ Experiment is known as a Trial & the
outcomes are known as Events or Cases.
 Throwing a die is a Trial & getting 1 (2,3,…,6) is an
event.
 Tossing a coin is a Trial & getting Head (H) or Tail
(T) is an event.

 A probability experiment is a chance process
that leads to well-defined results called
outcomes.

 An outcome is the result of a single trial of a
probability experiment.

 A sample space is the set of all possible
outcomes of a probability experiment.

 An event is the collection of one or more
outcomes of an experiment

Experiments, Events and Outcomes

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Assigning Probabilities
Three approaches to assigning
probabilities --
◦Classical
◦Empirical
◦Subjective
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Mathematical/ Classical/ „a priori‟ Probability

 Basic assumption of classical approach is that the
outcomes of a random experiment are “equally likely”.

 According to Laplace, a French Mathematician:
“Probability, is the ratio of the number of „favorable‟ cases
to the total number of equally likely cases”.

 If the probability of occurrence of A is denoted by
p(A), then by this definition, we have:

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Limitations of Classical definition
 Classical probability is often called a priori probability
because if one keeps using orderly examples of
unbiased dice, fair coin, etc. one can state the answer
in advance (a priori) without rolling a dice, tossing a
coin etc.

 Classical definition of probability is not very
satisfactory because of the following reasons:
◦ It fails when the number of possible outcomes of the
experiment is infinite.
◦ It is based on the cases which are “equally likely” and as
such cannot be applied to experiments where the
outcomes are not equally likely.

Relative/ Statistical/ Empirical Probability
 Empirical Probability of an event is an "estimate" that the
event will happen based on how often the event occurs after
collecting data or running an experiment (in a large number of
trials). It is based specifically on direct observations or
experiences.
 Empirical Probability Formula

 P(E) = probability that an event, E, will occur.
n(E) = number of equally likely outcomes of E.
n(S) = number of equally likely outcomes of sample
space S.

Limitations of Statistical/ Empirical
method
 The Empirical probability P(A) defined earlier can
never be obtained in practice and we can only attempt
at a close estimate of P(A) by making N sufficiently
large.

 The experimental conditions may not remain
essentially homogeneous and identical in a large
number of repetitions of the experiment.

 The relative frequency of m/N, may not attain a unique
value, no matter however large N may be.

Subjective Probability

 If there is little or no past experience or information on which to
base a probability, it may be arrived at subjectively.

 Illustrations of subjective probability are:
1. Estimating the likelihood Tiger Woods will win the Grand Slam in 2009.
2. Estimating the likelihood you will become a millionaire by 2015.
3. Probability President Obama will win the 2012 Presidential election.

Summary of Types of
Probability

Glossary of terms
 Classical Probability: It is based on the idea that
certain occurrences are equally likely.
◦ Example: - Numbers 1, 2, 3, 4, 5, & 6 on a fair die
are each equally likely to occur.
 Conditional Probability: The probability that an event
occurs given the outcome of some other event.
 Independent Events: Events are independent if the
occurrence of one event does not affect the
occurrence of another event.
 Joint Probability: Is the likelihood that 2 or more
events will happen at the same time.
 Multiplication Formula: If there are m ways of doing
one thing and n ways of doing another thing, there are
m x n ways of doing both.

 Outcome: Observation or measurement of an
experiment.
 Prior Probability: The initial probability based
on the present level of information.
 Probability: A value between 0 and
1, inclusive, describing the relative possibility
(chance or likelihood) an event will occur.
 Subjective Probability: Synonym for personal
probability. Involves personal
judgment, information, intuition, & other
subjective evaluation criteria.
◦ Example: - A physician assessing the
probability of a patient‟s recovery is making a
personal judgment based on what they know
and feel about the situation.

Probability

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