# Business Statistics - Probability and Probability distribution by gauravjindal

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```									Probability and Probability distribution
Basic Concepts of Probability
• Learning Objectives
– Appreciate relevance of probability theory in
decision- making
– Understand different approaches to probability
– Calculate probabilities in different situations
– Revise probability estimate , if added info is
available
Relevance of Concept of Probability
• Uncertainty.
– Weather, stock market prices, product quality,
• Decision- making in such areas is facilitated
through formal and precise expressions for the
uncertainties involved
– Help towards taking right decisions.
Basic Concepts
• Probability in common parlance connotes the chance
of occurrence of an event or happening
• In order that we measure it a formal definition is
required
• This is achieved through the study of certain basic
concepts in probability theory, like:
– Experiment
– Sample size
– and event
Experiment
• Any action (tossing a coin, measurement to assess
quality, launching a product) constitute an
experiment in probability theory terminology
• Experiments have thre things in common
– There are two or more outcomes of each experiment
– It is possible to specify the outcomes in advance
– there is uncertainty about the outcomes
Sample size
• The set of all possible outcomes of an
experiment is defined as a sample size
• Given an experiment, the sample space is
fully determined by listing down all the
possible outcomes of an experiment.
event
• One or more possible outcomes of an experiment
• Thus an event is a subset of sample space.
• Refer to a single outcome or a combination of
outcomes.,
– In the case of a market study experiment of a product:
Demand for the product is uncertain and hence let us take
51sample points constituting a sample size.
– We may obtain: The event that demand is exactly 100,
The event that demand is 101 or 120, The event that
demand lies between 101 to 102
3 Different approaches to probability
theory (mainly to cater to the 3 different
types of situations under which probability
measures are usually sought
• We may explore approaches through:
• Examples of distinct types of experiments
• The axioms common to these approaches are then
presented
• Concept of probability defined using the axioms.
3 distinct types of experiments and events
interested in
• Situation1.:
– Experiment: Drawing a number from among nine
numbers (say 1 to 9)
– Event: On any draw , number 4 occurs
• Outcome Marked by the presence of “equality likely”
• Classical approach to probability theory: can be defined as the
number of outcomes favorable to the event, divided by total
number of outcomes.
• Thus, P(A)=1/9 where in the event “a 4 comes out in a draw”
is denoted by A, and the probability of the event as, P(A)
Ctd---------
• Situation 2:
– Experiment : administers a particular drug
– Event : The drug puts a person to sleep in 10 minutes
• We find it difficult to apply the earlier definition of probability
here as: we are not sure whether the drug would be equally
effective to all, and also how many are tested.
• The Relative Frequency Approach is used to compute in such
cases. Probability of occurrence of an event is given by the
ratio of the number of times the event occurs to the total
number of trials.
• Denoting the event by P(B), we can write: P(B)=no. of
persons who fell sleep in 10 minutes, divided by total no. of
persons given drug.
• In order to take such measure, total number of trials in the
experiment should be large.
CTD---------
• Situation 3:
– Experiment: commissioning a solar power plant
– Event : The plant turns out to be successful
• The situation seems Apparently similar to second one, and
hence may be tempted to use Relative frequency Approach ,
and P(C)=no. of successful ventures divided by total number of
such ventures taken up.
• However, the calculation presupposes: it is possible to do an
experiment with such venture, and past data on such ventures
undertaken are available.
• Here past data, experimentation are ruled out, and hence only
way out is “Subjective Approach to Probability”
• In this approach we try to assess the probability from our own
experience.
Axioms fundamental to probability theory,
providing unified approach to probability
• The probability of an event P(A), must be a number
between 1 and 0
• The probability of occurrence of one of the other of
all possible events,P(S)=1
• If two events are such that occurrence of one implies
that the other cannot occur , then the probability
that either one or the other will occur is equal to the
sum of their individual probabilities. These events
are called mutually exclusive events.
Definition of Probability
• Given the axioms, we may define probability as a
function which assigns probability value P to each
sample point of an experiment abiding by the
axioms.
• Thus axioms themselves define probability.

• P10(ex-3,4)
Venn Diagram and solutions
Bayes’ theorem and Revising
Probability Estimate
• Decision making---an ongoing process
• The revision of probability with added information
is formalized in probability theory in terms of
Bayes’ theorem
• Bayes’s Theorem:
– If A and B are two mutually exclusive and collectively
exhaustive events and C is another event defined in the
context of same experiment, then conditional probability
given
Discrete Probability Distribution
• Learning Objectives:
– Understand the concept of random variable and
probability distribution
– Usefulness of probability distribution in decision making
– Identify situations where discrete probability distribution
can be applied.
– application of summary measures of a discrete
probability distribution
Basic Concepts
Random variable and Probability
Distribution
• The uncertain real variable assumes different
numerical values depending on the outcome of an
experiment. And to each of these values a
Probability assignment can be made, and is known
as Random Variable
• The resulting representation of all the values with
their probabilities is termed the Probability
Distribution (of the uncertain real value)
Discrete Random Variable

• Where the uncertain real value takes only discrete
value the variable is called discrete random variable
and the resulting distribution is a discrete probability
distribution.
Continuous random Variables
• The types of random variables which can
take infinitely large number of values is
called Continuous random Variables
• The resulting distribution is called
Continuous Probability distribution
Discrete probability Distribution
• In discrete situations, the function that gives
the probability of every possible outcome is
referred to in Probability theory as the
“Probability mass function” (p.m.f)
• Assessing p.m.f of a random variable
follows directly from what we saw earlier.
The different methods by which p.m.f of a
random variable can be specified are using:

• Std functions in probability theory
• Past data on random variable
• Using subjective assessment

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