Intro to Probabiltiy theory notes for Elements Class.pptx

Symbolically pe prob of e pe m n 16 some basic

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Symbolically- P(E) – Prob. of E P(E)= m N
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16 Some Basic Probability Concepts Contd.. Relative frequency probability If the process is repeated a large number of times, n , and if some resulting event with the characteristic E occurs ‘m’ times, the relative frequency of occurrence of E, m/n, will be approximately equal to the probability of E. Symbolically P(E) = m/n
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Subjective Probability (In the early 1950s, L.J. Savage) Probability measures the confidence that a particular individual has in the truth of a particular proposition. This concept doesn’t rely on the repeatability of any process. In fact, by applying this concept of probability, one may evaluate the probability of an event that can only happen once, for example, the probability that a cure for Ebola will be discovered within the next 10 years 17
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18 Some Basic Probability Concepts Contd.. ELEMENTARY PROPERTIES OF PROBABILITY (Axiomatic approach to probability ( A. N. Kolmogorov in 1933 ). The basics of this approach is embodied in three properties: Axiom of non-negativity : Given some process (or experiment) with n mutually exclusive outcomes (called events) - E 1 , E 2 , ……E n , the probability of any event E i is assigned a nonnegative number. That is P(E i )≥ 0
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19 Some Basic Probability Concepts Contd.. Axiom of exhaustiveness : The sum of the probabilities of the all mutually exclusive and exhaustive outcomes is equal to 1. i.e.P(E 1 )+ P (E 2 ) +………P(E n ) = 1. This is the property of exhaustiveness. Axiom of additiveness : Consider any two mutually exclusive event, E i and E j . The probability of the occurrence of either E i or E j is equal to the sum of their individual probabilities. P(E i or E j ) = P(E i )+ P(E j ) Suppose the two events were not mutually exclusive ; in that case the probability: P(E i or E j ) = P(E i )+ P(E j ) – P(E i ∩ E j )
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Some Basic Probability Concepts Contd.. (Normalization) The probability of the entire sample space Ω is equal to 1, that is, P(Ω) = 1 . 20
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21 Definition of Probability When all the equally possible occurrences have been enumerated, the probability ( Pr) of an event happening is the ratio of the number of ways in which the particular event may occur to the total number of possible occurrences.
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Probability Statements Suppose the Pr of a machine producing a defective item is known then the Pr that in a random sample of eight items produced by the machine, the Pr of not more than 2 being defective will be given by the probability statement: Pr(not more than 2 being defective) = P(≤2) = P(0) + P(1) + P(2) i.e. P(no defects) + P(1 defect) + P(2 defects) 22
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23 Again the chance or probability of say at least 3 items being defective would be given by: P(≥3) = P(3) + P(4) + . . . + P(8) - - - - (A) Or since the total Pr = 1 we could write (A) as: P(≥3) = 1 – [P(0) + P(1) + P(2)] Note: This is a very useful way of writing (A) and one we will be using regularly in our work.
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