INTRODUCTION to PROBABILITY

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INTRODUCTION to PROBABILITY
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BASIC CONCEPTS of PROBABILITY

• Experiment
• Outcome
• Sample Space
• Discrete
• Continuous
• Event

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Interpretations of Probability

• Mathematical
• Empirical
• Subjective

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MATHEMATICAL PROBABILITY

• P(E)

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PROPERTIES

• 0 lt P(E) lt 1
• P(E) 1 - P(E)
• P(A or B) P(A) P(B)
• for two events, A and B, that do not intersect

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Example

• A part is selected for testing. It could have
  been produced on any one of five cutting tools.
• What is the probability that it was produced by
  the second tool?
• What is the probability that it was produced by
  the second or third tool?
• What is the probability that it was not produced
  by the second tool?

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INDEPENDENT EVENTS

• Events A and B are independent events if the
  occurrence of A does not affect the probability
  of the occurrence of B.
• If A and B are independent
• P(A and B) P(A)P(B)

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Example

• The probability that a lab specimen is
  contaminated is 0.05. Two samples are checked.
• What is the probability that both are
  contaminated?
• What is the probability that neither is
  contaminated?

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DEPENDENT EVENTS

• Events A and B are dependent events if they are
  not independent.
• If A and B are independent
• P(A and B) P(A)P(B/A)

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Example

• From a batch of 50 parts produced from a
  manufacturing run, two are selected at random
  without replacement?
• What is the probability that the second part is
  defective given that the first part is defective?

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MUTUALLY EXCLUSIVE EVENTS

• Events A and B are mutually exclusive if they
  cannot occur concurrently.
• If A and B are mutually exclusive,
• P(A or B) P(A) P(B)

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NON MUTUALLY EXCLUSIVE EVENTS

• If A and B are not mutually exclusive,
• P(A or B) P(A) P(B) - P(A and B)

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Example

• Disks of polycarbonate plastic from a supplier
  are analyzed for scratch resistance and shock
  resistance. For a disk selected at random, what
  is the probability that it is high in shock or
  scratch resistance?
• Shock Resistance
• high low
• Scratch R high 80 9
• low 6 5

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RANDOM VARIABLES

• Discrete
• Continuous

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DISCRETE RANDOM VARIABLES

• Maps the outcomes of an experiment to real
  numbers
• The outcomes of the experiment are countable.
• Examples
• Equipment Failures in a One Month Period
• Number of Defective Castings

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CONTINUOUS RANDOM VARIABLE

• Possible outcomes of the experiment are
  represented by a continuous interval of numbers
• Examples
• force required to break a certain tensile
  specimen
• volume of a container
• dimensions of a part

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Discrete RV Example

• A part is selected for testing. It could have
  been produced on any one of five cutting tools.
  The experiment is to select one part.
• Define a random variable for the experiment.
• Construct the probability distribution.
• Construct a cumulative probability distribution.

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EXPECTED VALUE

• Discrete Random Variable
• E(X) X1P(X1) . XnP(Xn)

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Example

• At a carnival, a game consists of rolling a fair
  die. You must play 4 to play this game. You roll
  one fair die, and win the amount showing (e.g...
  if you roll a one, you win one dollar.) If you
  were to play this game many times, what would be
  your expected winnings? Is this a fair game?

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CUMULATIVE PROBABILITY FUNCTIONS

• For a discrete random variable X,
• the cumulative function is
• F(X) P(X lt x)
• S f(z) for all z lt x

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PROBABILITY HISTOGRAMS
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Variance of a Discrete Probability Distribution

• Var(X) Sx - E(X)2f(x)

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SOME SPECIAL DISCRETE RVs

• Binomial
• Poisson
• Geometric
• Hypergeometric

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BINOMIAL

• X the number of successes in n independent
  Bernoulli trials of an experiment
• f(x) nCxpx(1-p)n-x for x
  0,1,2.n
• f(x) 0 otherwise

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EXAMPLE

• A manufacturer claims only 10 of his machines
  require repair within one year.
• If 5 of 20 machines require repair, does this
  support or refute his claim??

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POISSON DISTRIBUTION

• X of success in an interval of time, space,
  distance
• f(x) e-l lx/x! for x 0,1,2,...
• f(x) 0 otherwise

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EXAMPLES

• Examples of the Poisson
• number of messages arriving for routing through a
  switching center in a communications network
• number of imperfections in a bolt of cloth
• number of arrivals at a retail outlet

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EXAMPLE of POISSON

• The inspection of tin plates produced by a
  continuous electrolytic process. Assume that the
  number of imperfections spotted per minute is
  0.2.
• Find the probability of no more than one
  imperfection in a minute.
• Find the probability of one imperfection in 3
  minutes.

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GEOMETRIC DISTRIBUTION

• X of trials until the first success
• f(x) px(1-p)n-x for x
  0,1,2.n
• f(x) 0 otherwise

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Example of Geometric

• The probability that a measuring device will show
  excessive drift is 0.05. A series of devices is
  tested. What is the probability that the 6th
  device will show excessive drift?
• Find the probability of the 1st drift on the 6th
  trail.
• P(X1) (0.05)(0.95)5 0.039