Bayes Theorem Partitions

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Bayes Theorem -- Partitions

• Given two events, R and S, if
• P(R ? S) 1
• P(R ? S) 0
• then we say that R and S partition the sample
  space.
• More than 2 events can partition the sample space

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Bayes Theorem

• Bayes theoremSuppose that B1, B2, B3,. . . ,
  Bn partition the outcomes of an experiment (the
  sample space) and that A is another event. For
  any number, k, with 1k n, we have the formula

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Example

• A Jar I contains 4 red balls and 2 blue balls and
  Jar II contains 3 red balls and 2 blue. The
  experiment is to chose a jar at random and from
  this jar select a ball and note the color of the
  ball.
• What is the probability that a blue ball is drawn
  from Jar II?

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Example 2

• All tractors made by a company are produced on
  one of three assembly lines, named Red, White,
  and Blue. The chances that a tractor will not
  start when it rolls off of a line are 6, 11,
  and 8 for lines Red, White, and Blue,
  respectively. 48 of the companys tractors are
  made on the Red line and 31 are made on the Blue
  line.
• What fraction of the companys tractors that do
  not start come from assembly line White?

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Example 3

• Thirty percent of the population have a certain
  disease. Of those that have the disease, 89
  will test positive for the disease. Of those
  that do not have the disease, 5 will test
  positive.
• What is the probability that a person has the
  disease, given that they test positive for the
  disease?

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Example 4

• A test attempts to recognize the presence of a
  certain disease. Records show that 10 of adults
  have a strong form of the disease, 18 have a
  mild form of the disease, and the rest have no
  form of the disease. A person with a strong form
  of the disease has a 15 chance of testing
  negative. A person with a mild form has a 10
  chance of testing negative. A person who does
  not have the disease has a 13 chance of testing
  positive, thereby falsely indicating that the
  person has the disease.

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Example 4 cont

• What is the probability that a person who has the
  disease will test positive?
• What is the probability that a person who is
  disease free will test negative?
• What is the probability that a person tests
  negative given that he does not the disease?

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Focus on the Project

• We would like to find P(S Y ? T ? C) and P(F
  Y ? T ? C) but couldnt find them directly
  because of the records
• We can, however, find P(Y ? T ? C S) and P(Y ?
  T ? C F)
• Now that we know Bayes Theorem we can find it
  indirectly

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Focus on the Project

• How can we use the two conditional probabilites
  that we can find to help us get what we want?
• Well, using Bayes Theorem we have the following

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Focus on the Project

• We have a similar formula for the probability of
  failure given the 3 conditions
• What is our last step to finding the probability
  that we want?

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Focus on the Project

• Now that we have our probabilites, we can
  cacluate the expected value and make a
  determination of foreclosure or workout
• We are not done yet!
• Do the same exact calculations but look at a
  range of years instead of one specific year