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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