Probability

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Probability

• Toolbox of Probability Rules

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Event

• An event is the result of an observation or
  experiment, or the description of some potential
  outcome.
• Denoted by uppercase letters A, B, C,

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Examples Events

• A Event President Clinton is impeached from
  office.
• B Event PSU mens basketball team gets lucky
  and wins their next game.
• C Event that a fraternity is raided next
  weekend.

Notation The probability that an event A will
occur is denoted as P(A).
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Tool 1

• The complement of an event A, denoted AC, is the
  event that A does not happen.
• P(AC) 1 - P(A)

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Example Tool 1

• Assume 1 of population is alcoholic.
• Let A event randomly selected person is
  alcoholic.
• Then AC event randomly selected person is not
  alcoholic.
• P(AC) 1 - 0.01 0.99
• That is, 99 of population is not alcoholic.

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Prelude to Tool 2

• The intersection of two events A and B, denoted
  A and B, is the event that both A and B
  happen.
• Two events are independent if the events do not
  influence each other. That is, if event A
  occurs, it does not affect chances of B
  occurring, and vice versa.

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Example for Prelude to Tool 2

• Let A event student passes this course
• Let B event student gives blood today.
• The intersection of the events, A and B, is the
  event that the student passes this course and the
  student gives blood today.
• Do you think it is OK to assume that A and B are
  independent?

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Example for Prelude to Tool 2

• Let A event student passes this course
• Let B event student tries to pass this course
• The intersection of the events, A and B, is the
  event that the student passes this course and the
  student tries to pass this course.
• Do you think it is OK to assume that A and B are
  not independent, that is dependent?

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Tool 2

• If two events are independent, then P(A
  and B) P(A) ? P(B).
• If P(A and B) P(A) ? P(B), then the two events
  A and B are independent.

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Example Tool 2

• Let A event randomly selected student owns
  bike. P(A) 0.36
• Let B event randomly selected student has
  significant other. P(B) 0.45
• Assuming bike ownership is independent of having
  SO P(A and B) 0.36 0.45 0.16
• 16 of students own bike and have SO.

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Example Tool 2

• Let A event randomly selected student is male.
  P(A) 0.50
• Let B event randomly selected student is sleep
  deprived. P(B) 0.60
• A and B randomly selected student is sleep
  deprived and male. P(A and B) 0.30
• P(A) P(B) 0.50 0.60 0.30
• P(A and B) P(A) P(B). So, being male and
  being sleep-deprived are independent.

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Prelude to Tool 3

• The union of two events A and B, denoted A or B,
  is the event that either A happens or B happens,
  or both A and B happen.
• Two events that cannot happen at the same time
  are called mutually exclusive events.

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Example to Prelude to Tool 3

• Let A event randomly selected student is drunk.
• Let B event randomly selected student is sober.
• A or B event randomly selected student is
  either drunk or sober.
• Are A and B mutually exclusive?

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Example to Prelude to Tool 3

• Let A event randomly selected student is drunk
• Let B event randomly selected student is in
  love
• A or B event randomly selected student is
  either drunk or in love
• Are A and B mutually exclusive?

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Tool 3

• If two events are mutually exclusive, then P(A
  or B) P(A) P(B).
• If two events are not mutually exclusive, then
  P(A or B) P(A)P(B)-P(A and B).

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Example Tool 3

• Let A randomly selected student has two blue
  eyes. P(A) 0.32
• Let B randomly selected student has two brown
  eyes. P(B) 0.38
• P(A or B) 0.32 0.38 0.70

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Example Tool 3

• Let A event randomly selected student does not
  abstain from alcohol. P(A) 0.75
• Let B event randomly selected student ever
  tried marijuana. P(B) 0.38
• A and B event randomly selected student drinks
  alcohol and has tried marijuana.
• P(A and B) 0.37
• P(A or B) 0.75 0.38 - 0.37 0.76

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Tool 4

• The conditional probability of event B given A
  has already occurred, denoted P(BA), is the
  probability that B will occur given that A has
  already occurred.
• P(BA) P(A and B) ? P(A)
• P(AB) P(A and B) ? P(B)

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Example Tool 4

• Let A event randomly selected student owns
  bike, and B event randomly selected student has
  significant other.
• P(BA) is the probability that a randomly
  selected student has a significant other given
  (or if) he/she owns a bike.
• P(AB) is the probability that a randomly
  selected student owns a bike given he/she has a
  significant other.

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Example Tool 4

• Let A event randomly selected student owns
  bike. P(A) 0.36
• Let B event randomly selected student has
  significant other. P(B) 0.45
• P(A and B) 0.17
• P(BA) 0.17 0.36 0.47
• P(AB) 0.17 0.45 0.38

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Tool 5

• Alternative definition of independence
• two events are independent if and only if P(AB)
  P(A) and P(BA) P(B).
• That is, if two events are independent, then
  P(AB) P(A) and P(BA) P(B).
• And, if P(AB) P(A) and P(BA) P(B), then A
  and B are independent.

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Example Tool 5

• Let A event student is female
• Let B event student abstains from alcohol
• P(A) 0.50 and P(B) 0.12
• P(AB) 0.50 and P(BA) 0.12
• Are events A and B independent?

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Example Tool 5

• Let A event student is female
• Let B event student dyed hair
• P(A) 0.50 and P(B) 0.40
• P(AB) 0.65 and P(BA) 0.52
• Are events A and B independent?