The document discusses key concepts in probability theory including probability, random experiments, sample spaces, events, random variables, probability distributions, and Bayes' theorem. It covers the binomial, Poisson, and normal distributions and their characteristics and applications. Decision theory is introduced as analyzing choices under uncertainty involving defining problems, identifying outcomes, assessing criteria, and evaluating alternatives to make optimal decisions.

1. PROBABILITY THEORY – THEORY OF
PROBABILITY , ADDITION AND
MULTIPLICATION LAW, BAYE’S
THEOREM

2. PROBABILITY
• Probability means possibility.
• It is a branch of mathematics that deals with the occurrence of a random event. The value is
expressed from zero to one.
• Probability has been introduced in Maths to predict how likely events are to happen.
• The meaning of probability is basically the extent to which something is likely to happen.
• This is the basic probability theory, which is also used in the probability distribution, where you will
learn the possibility of outcomes for a random experiment. To find the probability of a single event
to occur, first, we should know the total number of possible outcomes.

3. THEORY OF PROBABILITY
• The basic concept in probability theory is a random experiment which is an action with an
uncertain outcome
Eg : the outcome of a coin flip or roll of a die
• A probability is a numerical measure of the likelihood of a particular outcome of a random
experiment
• Probability theory makes use of some fundamentals such as sample space, probability
distributions, random variables, etc. to find the likelihood of occurrence of an event.

4. BASIC TERMINOLOGIES IN PROBABILITY
THEOREM
Random Experiment
A random experiment, in probability theory, can be defined as a trial that is repeated multiple times in order to
get a well-defined set of possible outcomes. Tossing a coin is an example of a random experiment.
Sample Space
Sample space can be defined as the set of all possible outcomes that result from conducting a random
experiment. For example, the sample space of tossing a fair coin is {heads, tails}.
Event
Probability theory defines an event as a set of outcomes of an experiment that forms a subset of the sample
space
TYPES OF EVENTS
Independent events
Dependent events
Mutually exclusive events
Equally likely events
Exhaustive events

5. Random Variable
In probability theory, a random variable can be defined as a variable that assumes the value of all
possible outcomes of an experiment. There are two types of random variables as given below:
• Discrete Random Variable
• Continuous random variable
Conditional Probability
When the likelihood of occurrence of an event needs to be determined given that another event has
already taken place, it is known as conditional probability. It is denoted as P(A | B).
Expectation
The expectation of a random variable, X, can be defined as the average value of the outcomes of an
experiment when it is conducted multiple times. It is denoted as E[X]. It is also known as
the mean of the random variable.
Variance
Variance is the measure of dispersion that shows how the distribution of a random variable varies
with respect to the mean. It can be defined as the average of the squared differences from the mean
of the random variable. Variance can be denoted as Var[X].

6. ADDITION LAW
The probability of occurrence of either event A or event B of two mutually exclusive (or
disjoint sets) events is equal to sum of their individual probabilities.
P (A ∪ B) = P (A) + P (B)

7. MULTIPLICATION LAW
The probability f the joint occurrence of event A and event B is equal to the conditional
probability of A given B, times the Probability of B
If A and B are dependent events
P(A ∩ B) = P(B) . P(A|B)
If A and B are two independent events
P(A ∩ B) = P(A) . P(B)

8. BAYE’S THEOREM
• The theorem is named after English statistician, Thomas Bayes, who discovered the formula
in 1763.
• The Bayes’ theorem is a mathematical formula used to determine the conditional probability
of events.
• The Bayes’ theorem describes the probability of an event based on prior knowledge of the
conditions that might be relevant to the event.

9. PROBABILITY THEORITICAL
DISTRIBUTION
 Concept and application of binomial distribution
 Concept and application of poisson distribution
 Concept and application of normal distribution

10.  The binomial distribution is the discrete probability distribution that gives only
two possible results in an experiment, either Success or Failure..
 Example :-
If we toss a coin, there could be only two possible outcomes:
heads or tails,
and if any test is taken, then there could be only two results:
pass or fail.
This distribution is also called a binomial probability distribution.
BINOMIAL DISTRIBUTION

11.  q = (1 - p)
probability of success, p, remains constant from trial to trial (so is the probability of failure
q , where q = (1-p)
Binomial Distribution Formula
The binomial distribution formula is for any random variable X, given by;
P(x:n,p) = nCx px (1-p)n-x
Or
P(x:n,p) = nCx px (q)n-x
Where,
n = the number of experiments
x = 0, 1, 2, 3, 4, …
p = Probability of Success in a single experiment
q = Probability of Failure in a single experiment = 1 – p
The binomial distribution formula can also be written in the form of n-Bernoulli
trials, where nCx = n!/x!(n-x)!. Hence,
P(x:n,p) = n!/[x!(n-x)!].px.(q)n-x

12. Characteristics of binomial distribution
 It is a discrete distribution which gives the theoretical probabilities
 For each trial there are only two possible outcomes on each trial, success (p) or a
failure (r).
 Each trial is independent and therefore the probability of success and the
probability of failure is the same for each trial.

13. Binomial Distribution Applications
 As we already know, binomial distribution gives the possibility of a different set
of outcomes. In real life, the concept is used for:
 Finding the quantity of raw and used materials while making a product.
 Taking a survey of positive and negative reviews from the public for any specific
product or place.
 By using the YES/ NO survey, we can check whether the number of persons
views the particular channel.
 To find the number of male and female employees in an organisation.
 The number of votes collected by a candidate in an election is counted based on 0
or 1 probability.

14. Poisson Distribution
 Poisson distribution is actually another probability distribution formula. As per
binomial distribution, we won’t be given the number of trials or the probability of
success on a certain trail. The average number of successes will be given in a
certain time interval.
 The average number of successes is called “Lambda” and denoted by the symbol
“λ”.

15. Poisson Distribution Formula

Here,is the average number
x is a Poisson random variable.
e is the base of logarithm and e = 2.71828 (approx).

16. APPLICATIONS OF POISSON
DISTRUBUTION
 To predict or explain the number of events occurring within a given interval of
time or space. “Events” could be anything from disease cases to customer
purchases to meteor strikes. The interval can be any specific amount of time or
space, such as 10 days or 5 square inches.
 Individual events happen at random and independently. That is, the probability of
one event doesn’t affect the probability of another event.
 You know the mean number of events occurring within a given interval of time or
space. This number is called λ (lambda), and it is assumed to be constant.

17. NORMAL DISTRIBUTION
 Normal distribution, also known as the Gaussian distribution, is a probability
distribution that is symmetric about the mean, showing that data near the mean are
more frequent in occurrence than data far from the mean

18. The Formula for the Normal Distribution
where:
x = value of the variable or data being examined and f(x) the probability function
μ = the mean
σ = the standard deviation

19. APPLICATIONS OF NORMAL
DISTRIBUTION
 The normal distributions are closely associated with many things such as:
 Marks scored on the test
 Heights of different persons
 Size of objects produced by the machine
 Blood pressure and so on.

20. DECISION THEORY
 INTODUCTION
 STEPS IN DECISION THEORY APPROACH
 TYPES OF DECISION ENVIRONMENT
 DECISION WORKING UNDER UNCERTAINITY & RISK

21. INTRODUCTION
Decision theory refers to a range of econometric and statistical tools for
analyzing an individual’s choices. In other words, it lets the entity make the
best logical decision possible when dealing with uncertain and unknown
circumstances. Analysts call it a theory of choice, relying on beliefs, attitudes,
and desires.

22.  There are three branches of decision theory:
 Normative decision theory: Concerned with the identification of optimal decisions, where
optimality is often determined by considering an ideal decision-maker who is able to
calculate with perfect accuracy and is in some sense fully rational.
 Prescriptive decision theory: Concerned with describing observed behaviors through the
use of conceptual models, under the assumption that those making the decisions are
behaving under some consistent rules.
 Descriptive decision theory: Analyzes how individuals actually make the decisions that
they do.

23. TYPES OF DECISION ENVIRONMENT

24.  1. Decision-Making Under Certainty
Coming on top, we have certainty. This is often regarded as the safest decision-making
environment and, consequently, the least risky. All the data you need is readily available, or
even if you need to gather it, it won’t be much expensive to get.
 2. Decision-Making Under Uncertainty
Uncertainty is when a manager fails to get enough information needed to make a sound
decision, or even if he does, it’s corrupted, biased, or has unreliable sources.
 3. Decision-Making Under Risk
Risk environments are when the probability of multiple events is tied to a decision. You’re
never sure about the outcomes of your decision other than calculated guesses. Such decisions
are associated with events that could either be very successful or quite disastrous for the
organization.

25. STEPS IN DECISION THEORY APPROACH
 1.Define the problem: Clearly state the problem and determine the decision variables,
uncertainties, and objectives.
 2.Identify the possible outcomes: List all the possible outcomes and their probabilities.
 3.Assess the decision criteria: Determine the consequences of each possible outcome, including
the costs and benefits.
 4.Evaluate the alternatives: Evaluate the alternatives by calculating the expected values of each
decision, taking into account the probabilities and consequences of each outcome.
 5.Select the best decision: Select the alternative with the highest expected value or the
alternative that maximizes the objective.
 6.Implement the decision: Take action to implement the selected alternative.
 7.Evaluate the results: Evaluate the results of the decision by comparing the actual outcomes
with the expected outcomes.