4203 Mathematical Probability Chapter 2: Probability

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4203 Mathematical ProbabilityChapter 2
Probability

• Instructor Dr. Ayona Chatterjee
• Spring 2011

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Classical Probability concept

• If there are N equally likely possibilities of
  which one must occur and n are regarded as
  favorable, or as a success, then the probability
  of success is given by the ration n/N.
• Example Probability of drawing a red card from a
  pack of cards 26/52 0.5.

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Probability

• Measures the likeliness of an outcome of an
  experiment.
• The probability of an event is the proportion of
  the time that events of the same kind will occur
  in the long run.
• Is a number between 0 and 1, with 0 and 1
  included.
• Probability of an event A is the ratio of
  favorable outcomes to event A to the total
  possible outcomes of that experiment.

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Some Definitions

• An experiment is the process by which an
  observation or measurement is made.
• When an experiment is preformed it can result in
  one or more outcomes which are called events.
• A sample space associated with an experiment is
  the set containing all possible outcomes of that
  experiment. It is denoted by S.
• Elements of the sample space are called sample
  points.

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Sample spaces

• Can be
• Finite (toss a coin)
• Countable (toss a coin till the first head
  appears)
• Discrete (toss a coin)
• Continuous (weight of a coin)

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Example

• A manufacturer has five seemingly identical
  computer terminals available for shipping.
  Unknown to her, two of the five are defective. A
  particular order calls for two of the terminals
  and is filled by randomly selecting two of the
  five that are available.
• List the sample space for this experiment.
• Let A denote the event that the order is filled
  with two nondefective terminals. List the sample
  points in A.

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Venn Diagram

• Graphical representation of sets.
• A is complement of event A.

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Probability of an Event

• POSTULATE 1 The probability of an event is a
  nonnegative real number that is P(A)gt0 for any
  subset A of S.
• POSTULATE 2 P(S) 1.
• POSTULATE 3 If A1, A2, .is a finite or infinite
  sequence of mutually exclusive events of S then
  P(A1 U A2 U..)P(A1 )P(A2 )..

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Mutually exclusive events

• Two events A and B are said to be mutually
  exclusive if there are no elements common to the
  two sets. The intersection of the two sets is
  empty.

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Rules of Probability

• If A and A are complementary events in a sample
  space S, then P(A)1-P(A). Thus P(S) 1.
• P(F) 0, probability of an empty set 0.
• If A and B are events in a sample space S and A
  is subset B the P(A) lt P(B).
• If A and B are any two events in a sample space
  then P(A U B) P(A)P(B) P(A B).

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Conditional Probability

• The conditional probability of an event A, given
  that an event B has occurred, is equal to
• Provided P(B) gt 0.
• Theorem If A and B are any two events in a
  sample space S and P(A)?0, then

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Independent Events

• Two events A and B are independent if and only if
  
• If A and B are independent then A and B are also
  independent.
• 
  
  
  
  

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Example

• A coin is tossed three times. If A is the event
  that a head occurs on each of the first two
  tosses, B is the event that a tail occurs on the
  thrid toss and C is the event that exactly two
  tails occur in the three tosses, show that
• Events A and B are independent.
• Events B and C are dependent.

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Bayes Theorem

• If B1, B2, ., and Bk constitute a partition of
  the sample space S and P(Bi) ?0, for i1, 2, .,
  k, then for any event A in S such that P(A) ?0

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Reliability of a product

• The reliability of a product is the probability
  that it will function within specified limits for
  a specific period of time under specified
  environmental conditions.
• The reliability of a series system consisting of
  n independent components is given by
• The reliability of a parallel system consisting
  of n independent components is given by
• Ri is the reliability of the ith component.