The document provides a comprehensive overview of probability, discussing its definition, methods of assigning probabilities (classical, relative frequency, and subjective methods), and key concepts such as experiments, sample spaces, and types of events (mutually exclusive, independent, etc.). It also covers probability laws, including the addition and multiplication laws, conditional probabilities, and Bayes' rule. The document serves as a foundational resource for understanding probability and its applications.

1. PROBABILITY
Pooja Yadav,
Research Scholar
Central University of Rajasthan
Satyender Yadav
Research Scholar,
UCCMS, MLSU, Udaipur

2. Topics to be covered

3. PROBABILITY
• Probability is the measure of the likelihood that an event will occur in a
Random Experiment.
• Probability is quantified as a number between 0 and 1. Where, “0”
indicates impossibility and “1” indicates certainty.
• The higher the probability of an event, the more likely it is that the event
will occur.

4. Methods of Assigning Probabilities
• The Classical Method
• The Relative Frequency of
occurrence method
• Subjective Probabilities
There are 3
general
methods of
assigning
probabilities:

5. The Classical Method
• When probabilities are assigned based on laws and rules, the method is
called as the Classical Method of assigning probabilities.
• Number of outcomes leading to the event divided by the total number of
outcomes possible. Each outcome is equally likely.
• Determined a priori -- before performing the experiment
• Applicable to games of chance
• Range of Possible Probabilities :- 0 ≤ P(E) ≤ 1

6. The classical method
• Formula
Where,
N = Total possible number of outcomes of an experiment
ne = The number of outcomes in which the event occurs out of N outcomes
N
EP ne
)(

7. RELATIVE FREQUENCY OF OCCURRENCE
• The probability of an event occurring is equal to the number of times the
event has occurred in the past divided by the total number of opportunities
for the event to have occurred.
• Relative frequency of occurrence is based on what has happened in the
past (Cumulated historical data).
P (E) = Number of Times an Event Occurred
Total Number of Opportunities for the Event to Occur

8. Subjective probability
• The subjective method of assigning probability is based on the feelings or
insights of the person determining the probability.
• It’s based on the person’s intuition or reasoning.
• Subjective probability is the accumulation of knowledge, understanding,
experience stored and processed in the human mind.
• Subjective probability can be used for unique (single trial) experiments
such as: - New product introduction
- Site selection decision
- Sporting Events

9. Structure of Probability
• It helps in developing a language of terms and symbol.
• The structure of probability provides a common framework within which
the topics of probability can be explored.

10. Structure of Probability
• Experiment :- An experiment is a process that produces outcomes.
Ex. :- Interviewing 20 randomly selected consumers and asking them which brand of
appliance they prefer.
• Sample Space :- A sample space is a complete roster or listing of all
elementary events for an experiment.
Ex. :- The sample space for an roll of a single die is {1,2,3,4,5,6}

11. Structure of Probability
• Unions :- The union of X,Y is formed by combining elements from each of the sets.
 Union is denoted as X ∪ Y.
 An element qualifies for the union of X,Y if it is in either X or Y or in both X and Y.
Example :- X = {1,4,7,9} and Y = {2,3,4,5,6}
X ∪ Y = {1,2,3,4,5,6,7,9}

12. Structure of Probability
• INTERSECTION :- The intersection contains the elements of common
to both sets.
 An intersection is denoted X ∩ Y.
 To qualify for intersection, an element must be in both X and Y.
 Example :- X = {1,4,7,9} and Y = {2,3,4,5,6}
X ∩ Y = {4}

13. Structure of Probability
• Event :- An event is an outcome of an experiment.
 The experiment defines the possibilities of the event
 Example :- In an experiment to roll a die, one event could be to roll an even number and
another event could be to roll a number greater than two.
• Elementary Events :- Events that cannot be decomposed or broken down into other events are
called elementary events.
 Elementary events are denoted by lowercase letters (e.g., e1, e2, e3…..)
 Example :- The experiment to roll a die, the elementary events for this experiment are to roll a
1 or roll a 2 or roll a 3, and so on.

14. Structure of Probability
• Mutually Exclusive Events :- Two or more events are mutually exclusive
events if the occurrence of one event precludes the occurrence of the other
event(s).
• Mutually exclusive events cannot occur simultaneously and therefore can have no
intersection.
• Probability of two mutually exclusive events occurring at the same time is zero.
• Example :- In the toss of a single coin, heads and tails are mutually exclusive events.
The person tossing the coin gets either a head or a tail but never both.

15. Structure of Probability
• Independents Events :- If the occurrence or nonoccurrence of one of the
events does not affect the occurrence or non occurrence of the other events
are called independent events.
 Example :- The event of getting a head on the first toss of a coin is independent of
getting a head on the second toss.

16. Structure of Probability
• Collectively Exhaustive Events :- Events which contains all possible
elementary events for an experiment are called collectively exhaustive
events.
• All sample spaces are collectively exhaustive lists.
• The sample space for an experiment can be described as a list of events that are
mutually exclusive and collectively exhaustive.

17. Structure of Probability
• Complementary Events :- The complement of event A is denoted A′,
pronounced as “not A”.
• All the elementary events of an experiment not in A comprise its complement.
• Example :- If event A is getting a 5 on the roll of a die, the complement of A is
getting a 1,2,3,4 or 6.
• Probability of the complement of A =
P (A′) = 1 – P(A)

18. Types of Probabilities
Marginal
The probability of
X occurring
Union
The probability of
X or Y occurring
Joint
The probability of
X and Y occurring
Conditional
The probability of
X occurring given
that Y has occurred
YX YX
Y
X
)(XP P X Y( ) P X Y( ) P X Y( | )

19. Laws of probability

20. The addition laws
• The general law of addition is used to find the probability of the union of
two events, P(X ∪ 𝑌).
• The expression P (X ∪ 𝑌) denotes the probability of X occurring or Y
occurring or both X and Y occurring.
• General law of Addition :- P (X ∪ 𝑌) = P(X) + P (Y) – P(X ∩ Y)
Where X,Y are events and (X ∩ Y) is the intersection of X and Y.

21. The addition laws
• Probability Matrix :- A probability matrix displays the marginal
probabilities and the intersection probabilities of a given problem.
• A probability matrix is constructed as a two-dimensional table with one
variable on each side of the table.
• Union probabilities and conditional probabilities must be computed from
the matrix.

22. Conditional Probability
• The conditional probability of (X│Y) is the probability that X will occur given Y
• The formula for conditional probability is derived by dividing both sides of the
general law of multiplication by P(Y).
• If X,Y are two events, the conditional probability of X occurring given that Y is
known or has occurred is expressed as P(X│Y).
• Law of Conditional Probability :- P(X│Y) = P (X ∩ Y) = P(X) . P (Y │ X)
P (Y) P(Y)

23. The multiplication laws
• The general law of multiplication is used to find the joint probability.
• The probability of intersection of two events (X ∩ Y) is called the joint
probability.
• Law of multiplication gives the probability that both event X and event Y
will occur at the same time.
• General Law of Multiplication :- P(X∩Y) = P(X)*P(Y│X) = P(Y)*P(X│Y)

24. Bayes’ rule
• It was developed by Thomas Bayes.
• Bayes’ rule is a formula that extends the use of the law of conditional
probabilities to allow revision of original probabilities with new
information.
• BAYES’ Rule :-
)()|()()|()()|(
)()|(
)|(
2211 nn
ii
i
XPXYPXPXYPXPXYP
XPXYP
YXP



25. Bayes’ rule
• The numerator of Bayes’ rule and the law of conditional probability are
the same – the intersection of Xi and Y shown in the form of the general
rule of multiplication.
• The denominator of Bayes’ rule includes a product expression
(intersection) for every partition in the sample space, Y, including the
event (Xi) itself.
• It represents a weighted average of the conditional probabilities, with the
weights being the prior probabilities of the corresponding event.

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