General Probability Rules

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General Probability Rules
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Review

• Probability of when E and F are disjoint is
  easy
• If E and F are independent then we can compute

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General Addition Rule for Two Events

• For any two events E and F,

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General Multiplication Rule for Two Events

• For any two events E and F,

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Example

• 60 of customers of an insurance agency have
  automobile policies, 40 have homeowners and 25
  have both. What is the probability that he or
  she has at least one of these two types? (Note a
  customer may have only a life insurance policy.)
• Let
• Eevent customer has auto insurance
• Fevent customer has homeowners

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Example

• P(customer has at least one of the two types of
  policy)

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Example

• P(customer has exactly one of the two types of
  policy)
• (Mr. Nollen Draws Venn Diagram)

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Example

• Suppose 20 of teenage drivers received a
  citation for a moving violation in the past year.
  Of those 80 went to driving school so the
  citation would not appear on their record. If a
  teenage driver is randomly selected, what is the
  probability that he or she received a citation
  and attended traffic school?

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Example (cont)

• LetFselected driver received a
  citationEselected driver attended traffic
  school
• What are we looking for?
• What do we know?

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Example (cont)

• What can we use to find
• Multiplication rule

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Example II

• The table gives information on VCRs sold by an
  appliance store
• What is the probability that the selected
  customer purchased a brand 1 model and an
  extended warranty?

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Example II (cont)

• Lets define the following events
• B1 event that brand 1 is purchased
• B2 event that brand 2 is purchased
• E Event that an extended warranty is purchased
• What is

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Example II (cont)

• Lets define the following events
• B1 event that brand 1 is purchased
• B2 event that brand 2 is purchased
• E Event that an extended warranty is purchased
• Find (Do it now)

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Example II (cont)

• Now, lets calculate P(E), the probability that
  an extended warranty is purchased.
• This can happen two different ways buy brand 1
  and warranty or buy brand 2 and warranty.
• Furthermore, since each customer purchased a
  single VCR, they could not have purchased both
  brand 1 and brand 2, so the two events are
  disjoint.

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Example II (cont)

• Finding P(E) (that an extended warranty is
  purchased.

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The Law of Total Probability

• IF B1 and B2 are disjoint events with
• P(B1)P(B2)1, then for any event E

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The Law of Total Probability

• 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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Last Example

• 68 of elementary school students from Beachwood
  said that they always wear a helmet when
  bicycling, but only 23 of students from Morland
  wear a Helmet.
• B selected student from Beachwood
• M selected student from Morland
• H select student always wears a helmet
• P(B) P(M) .5

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Last Example

• What portion of students always wears helmets?
  (Do it now.)
• B selected student from Beachwood
• M selected student from Morland
• H select student always wears a helmet
• P(B) P(M) .5
• 44.5 of the school children in these two
  communities always wear a helmet

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Practice

• DO 6.60 Now!
• Start by defining two events and then stop!
• Now, using those events, show everything the
  problem asserts.
• What is question (a) asking?
• Do it (a) now!
• What is part (b) asking?
• Do (b) now.
• What is question (c) asking?
• Do (c) now.
• Do (d) now.

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Homework

• 6.53-6.63 odd