Math 201

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Math 201

• 4.5 General Prob. Rules

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Last Time

• Probability distributions
• Finding probabilities for continuous rvs
• Uniform
• Normal
• Computing means and standard deviations for
  discrete rvs
• LLN

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Caution

• Its important to be able to distinguish between
• a distribution of data vs. a probability
  distribution
• sample mean x vs. m
• sample variance s2 vs. s2

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Probability Rules

• For any event A, 0 P(A) 1
• P(S) 1.
• Complement Rule
• P(Ac)1-P(A)
• Addition Rule If A and B are disjoint
• P(A or B )P(A)P(B)
• Multiplication Rule If A and B are independent
• P(A and B )P(A) P(B)

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General Addition Rules

• If A, B, and C are disjoint
• P(one or more of A, B, C) P(A)P(B)P(C)

B
A
C
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Examples

• Pick a student at random
• What is the probability that the student has an A
  or AB or B?

• Example draw a card from a deck
• What is probability that it is a
• Heart or Club or Spade?
• 2. What is probability that it is a Heart or King
  or Queen?

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General Addition Rule

• For any two events A and B
• P(A or B )P(A)P(B) - P(A and B)

Example draw a card from a deck What is
probability that it is a Heart or King ?
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Addition Rules

• If A and B are disjoint
• P(A or B )P(A)P(B)
• P(A or B )P(A)P(B) - P(A and B )
• Note If A and B are disjoint
• P(A and B) 0

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Example

• Pick a student a random
• What is the probability that the student is a
  Female?
• What is the probability that the student likes
  Coke?
• What is the probability that the student is a
  Female or likes Coke?

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Example

• Pick a student at random from the group.
• P(student has a B and has perfect attendance)?
• P(student has a B or has perfect attendance)?
• P(student has a passing grade and has perfect
  attendance)?
• P(student does not have perfect attendance)?
• P(student has a CD)?

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Professor Jackson Prepares Kids for a H.S.
Equivalency Exam.

• 80 need work in Math
• 70 need work in English
• 55 need work in both Math and English.
• a) Draw a Venn Diagram.  
• b) Are the events M and E disjoint?  
• c) Find the probability that a student selected
  at random needs help in at least one  of the two
  subjects.

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Conditional Probability

• Event A You will fall down when you walk to work
  today.
• Guess at Probability of A__________
• Event B The sidewalks are icy from freezing rain
  last night
• Guess at New Probability of A__________
• A,B independent?________________

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Conditional Probability

• Event A Man will land on the moon in the next 10
  years.
• Guess at probability of A___________
• Event B Scientists will soon discover a cure for
  the AIDS virus.
• New guess at probability of A___________
• A, B independent? ______________

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Conditional Probability

• Event A You will win the powerball today.
• Guess at probability of A___________

• Event B You did not buy a powerball ticket.
• New guess at probability of A___________
• A, B independent? ______________

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Conditional Probability

• P(likes coke)?
• If the student is female, what is the probability
  that the student likes coke?

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Example

• Pick a student at random from the group.
• Suppose the student has perfect attendance, what
  is the probability that student has an A?
• Suppose the student has an A, what is the
  probability that student has perfect attendance?
• P(A/Poor)
• P(C/Poor)
• P(Poor/C)

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Independence Again

• If A and B are independent
• The occurrence of A is not going affect the
  probability of B!
• So P(B/A) P(B)
• Examples

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Multiplication Rule

• Since
• For any two events A and B
• P(A and B) P(A). P(B/A)
• Examples
• 102

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Multiplication Rules

• If A and B are independent
• P(A and B )P(A) P(B)
• P(A and B )P(A) P(B/A)
• Note If A and B are indept P(B/A)P(B)

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Homework

• 4.87, 4.93, 4.95
• 4.89, 4.91