Probability

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Probability

• Bayes Theorem

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Bayes' Theorem (Two Events)
If B1 and B2 are mutually exclusive events, and
B1 ? B2 S, then for any event A in S,
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Bayes' Theorem (Three Events)
If B1, B2 and B3 are mutually exclusive events,
and B1 ? B2 ? B3 S, then for any event A in
S,
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Bayes' Theorem
If B1, B2,, Bn are mutually exclusive events,
and B1 ? B2 ?? Bn S, then for any event A in
S,
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Example Bayes' Theorem

• A printer has seven book-binding machines with
  the proportion of production and probability of
  defective binding listed in the following table.
  If a book is selected at random and found to have
  a defective binding, what is the probability that
  it was bound by machine 1?

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Example

• Ten percent of the pens made by Apex are
  defective. Only 5 of the pens made by its
  competitor, B-ink, are defective. Since Apex
  pens are cheaper than B-ink pens, an office
  orders 70 of its stock from Apex and 30 from
  B-ink. A pen is chosen at random and found to be
  defective. What is the probability that it was
  produced by Apex?

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Example

• A metropolitan police department maintains
  statistics of larcenies reported in the various
  precincts of the city. It records the proportion
  of the city population in each precinct and the
  precinct larceny rate ( the proportion of the
  precinct population reporting a larceny within
  the past year).
• A larceny victim is randomly chosen from the city
  population. What is the probability he or she
  comes from Precinct 3?