Bayes Rule

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Bayes Rule
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Bayes Rule

• Given P(EB1) how can we calculate P(B1E)?
• If B1 and B2 are disjoint events with
  P(B1)P(B2)1, then for any event E

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Bayes Rule

• More generically
• If B1, B2, BK are disjoint events with
  P(B1)P(B2)P(BK)1, then for any event E

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Example

• This data is taken from an article in American
  Journal of Clinical Pathology (1993) on
  Laboratory Considerations in the Diagnosis and
  Management of Lyme Borreloisis

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Example

• Notation
• represents a positive blood test
• - represents a negative blood test
• L event that a patient actually has Lyme
• LC event that a patient does not have Lyme

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Example

• Reported by the article
• P(L) .00207
• P(LC) .99793
• P(L) .937
• P(-L) .063
• P(LC) .03
• P(-LC) .97
• What is P(L)?

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Example

• What is P(L)?
• 1.56 of those who test positive actually have
  the disease

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Example

• Two shipping companies offer overnight delivery.
  A .com company ships 30 of its overnight
  packages using shipper 1 and 70 using service 2.
  Shipper 1 fails to meet the 10am delivery 10 of
  the time, and shipper 2 fails 8. Suppose that
  you made a purchase and requested overnight
  delivery, but it is late. Which shipping service
  is more likely to have been used?

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Example

• Events
• S1 event that package shipper was 1
• S2 event that package shipper was 2
• L event that package was late
• Whats known? (Do it now)
• What do we want? (Okay, do that now)

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Example

• Events
• S1 event that package shipper was 1
• S2 event that package shipper was 2
• L event that package was late
• Whats known? (Do it now)
• Now, calculate P(S1L)

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Example

• Events
• S1 event that package shipper was 1
• S2 event that package shipper was 2
• L event that package was late
• Whats known? (Do it now)
• Now, calculate P(S2L)

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Example

• So, which shipper was more likely to ship your
  package?
• Shipper 2.

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Homework

• 6.65 - 6.69