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Business Statistics - Probability and Probability distribution

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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
• head/tail, success/failures,
• 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