NA387 Lecture 6: Bayes Theorem, Independence

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NA387 Lecture 6 Bayes Theorem, Independence

• Devore, Sections 2.4 2.5

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Topics

• Conditional Probability
• Multiplication Rule
• Law of Total Probability
• Bayes Theorem
• Independence of Events

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Conditional Probability
Conditional Probability Relationships

• The intersection of two events may be re-written
  from the above using the multiplication rule.
• Multiplication rule is useful for determining
  probability of an event that depends on other
  events

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Multiplication Rule
U

• P(A B) P(B) P(AB) Or P(A) P(BA)
• Examples Car Crash Injuries
• Event A Getting injured in a crash (injuries
  hospital visits or fatalities)
• Event B US resident involved in a car crash
• P(B) 0.01
• P(A B) 0.30 -- Of those US residents
  involved in crashes, 30 get injured
• P (A B) What is the probability that a US
  resident will be in a car crash and get injured?
• P (getting injured and involved in a car crash)
  ?

U
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Conditional Probability and Tree Diagrams

• Suppose you produce three brands of TVs.
• Event A sell a TV (brands A1, A2, A3)
• Event B repair a sold TV
• Suppose Selling Mix A1 50, A2 30 and A3
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• Likelihood to Repair Given Model A1 25
• Likelihood to Repair Given Model A2 20
• Likelihood to Repair Given Model A3 10
• Draw a tree diagram and determine the probability
  that any unit will be repaired

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Conditional Probability and Venn Diagrams

• Event B repair sold TV B no repair
• Event A sell TV Brand (A1, A2, A3)

B
B
With multiple events forming an exhaustive set,
S, a general expression for total probability can
be developed.
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Law of Total Probability

• Given a collection of k mutually exclusive and
  exhaustive events (A1, .. Ak), for any event B
• Law of Total Probability
• P(B) P(BA1)P(A1) P(BAk)P(Ak)

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TV Example Revisited

• Using the law of total probability, compute the
  Probability that a TV will be repaired P(B).
• P(A1) 50 P(BA1) 25
• P(A2) 30 P(BA2) 20
• P(A3) 20 P(BA3) 10
• Find P(B) P(repair)
• Find P(B) P(Not repair)

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Bayes Theorem

• Using the law of total probability, we may
  develop a useful application of the
  multiplication rule
• given a set of prior probabilities P(Ai) (where
  P(Ai) gt 0) and the conditional probabilities P(B
  Ai), we may compute posterior probabilities
  provided P(B) gt 0
• Convert P(B Ai) to P(Aj B)

i prior j - posterior
Bayes Theorem
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Prior and Posterior Probability

• To appreciate Bayes Theorem, we must understand
  prior and posterior probability.
• Prior Probability initial probability -- e.g.,
  P(Ai)
• Often based on background data
• Posterior Probability updated probability
  arising from new information (i.e., condition --
  P(Aj B)).
• Often based on empirical evidence (i.e., sample
  observation)
• TV Example
• Prior Probability Prob A1 is sold P(A1) 50
• Posterior Probability Example What is the Prob
  (A1) if you know that the TV was repaired?
• P(A1 B) ?

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Prior and Posterior Probabilities

• So, why is Bayes Theorem so useful?
• Consider the following TV results
• Computing posterior probabilities allows us to
  update our decisions with new empirical
  information.

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Rare Disease Problem (Devore Example 2.30)

• Event A1 Have Disease
• Event A2 Do Not Have Disease
• Event B Positive Test Result for Disease
• What of tests yield positive results P(B)?
• What of positive test results actually will
  indicate the person has the disease?

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Sensitivity Analysis

• What happens if of false positives for the test
  reduces?
• What happens to the test results if the disease
  is not so rare?

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Independent Events

• If and only if (iff) A and B are independent
• If you have a set of mutually independent events
  (A1 .. An), then

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Independence Process Control Application

• Suppose you take samples of size 5 from a
  population to test for process stability. You
  assume your process has a normal distribution
  such that the mean 1000 mm.
• So, Event Ai sample i has a mean gt 1000
• What is the Prob (Ai)?
• What is the probability that three samples in a
  row will have a mean greater than 1000 (assume
  samples are independent)?
• If you get seven samples in a row, would you
  conclude the underlying mean of the population is
  still 1000, or would you conclude the mean has
  changed?

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Independence and Reliability Series Reliability

• Reliability probability that a system does not
  fail during a time interval (0 to t, or t1 to t2)
• Series Reliability (one unit fails, system fails)

R2
R1
i unit in series
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Independence and Reliability - Parallel

• Parallel System Reliability (if either unit
  survives, system survives)

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Series and Parallel Problem

• Twin Engine Example
• If both engines are independent and if P(Engine
  Survives) 0.99,
• What is the probability that both will survive?
• What is the probability that at least one will
  survive?
• What is the purpose of adding redundancy to a
  system (i.e., multiple components performing the
  same function)?
• Other Example alarm clock wake-up service

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Systems consisting of independent components

• Suppose you have a complex systems of
  independent components. To operate, component 3
  must survive and either component 1 or 2.
• Determine the overall system reliability (Rs) of
  this system if R1R2.95 and R3.80.

Rs?