STAT 200 Chapter 4 Sample Space

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STAT 200Chapter 4Sample Space Probability
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Lots of definitions . . .

• Probability experiment
• A chance process that leads to well-defined
  results called outcomes
• An Outcome
• Is the result of a single trial of an experiment
• Sample Space
• Is the set of all possible outcomes of a
  probability experiment
• An Event
• Consist of a set of outcomes of a probability
  experiment

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Experiments Sample Spaces

• Examples
• Flipping a coin H, T
• Rolling 1 die 1,2,3,4,5,6
• Rolling a pair of dice . . . table, p. 174
• More complex gender of children in sequence when
  family has three kids ?

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Gender of children in sequence when family has
three kids

• Use a tree diagram to help envision . . .
• Graphical device consisting of line segments from
  start point to outcome, used to organize display
  of ALL outcomes in probability experiment

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OK . . . What was an event again? - a set
of outcomes of a probability experiment
Event M those times when Markley Oil is
profitable Event C those times that Collins
Mining is profitable
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Bradley Investments did a 2-step experiment
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Interpretations of Probability

• Classical probability
• based on equally likely events
• Empirical probability
• Uses relative frequency/historical data
• Subjective probability
• probability assigned by expert using insight or
  intuition

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Classical probability
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Empirical probability
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Subjective probability
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Remember Bradley Investments?
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Note how outcomes and probabilities from previous
slide come together here
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Probability Rules

• The probability of an event E is a number between
  and including 0 and 1
• 0
  P(E) 1
• If an event E cannot occur its probability is 0
• If an event E is certain, its probability is 1
• The sum of the probabilities of all outcomes in
  the sample space must equal exactly 1

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Relationships in probabilityComplementary events

• The complement of an event E is the set of
  outcomes in the sample space that are not
  included in the outcomes of event E
• The complement of E is denoted by

Venn Diagram
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Relationships . . . Mutually Exclusive Events

• Two events are mutually exclusive events if they
  cannot occur at the same timethey have no
  outcomes in common.

Event B
Event A
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Addition Rules for Probability

• When two events A and B are mutually exclusive,
  the probability that A or B will occur is P(A or
  B) P(A) P(B)
• If A and B are not mutually exclusive, then P(A
  or B) P(A) P(B) P(A and B)

Addition rule Focus on the OR idea
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Multiplication Rules for Probability

• Two events A and B are independent events when
  the fact that A occurs does not affect the
  probability of B occurring
• When two events are independent, the probability
  of both occurring is
  P(A and B) P(A) P(B)
• When two events are dependent the probability of
  both occurring is P(A and B)
  P(A) P(BA)

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Conditional Probability

• The probability that the second event B occurs
  given that the first event A has occurred can be
  found this way

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Sec. 4-5 -- Counting Rules

• Fundamental counting rule
• In a sequence of events in which the first one
  has k1 possibilities and the second event has k2
  and the third has k3 and so forth, the total
  number of possibilities will be
• k1 k2 k3 ? ? ? kn
• Note In this case and means to multiply

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Factorial Notation Review

• See review in text Appendix A-1, p. A-1
• See Table A of factorials in AppC, p. A-17
• Factorial notation uses the exclamation point and
  involves multiplication
• 5! 54321 120
• 4! 4321 24
• 3! 321 12
• 2! 21 2
• 1! 1
• 0! 1

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Permutations

• A permutation is an arrangement of n objects in a
  specific order.
• The arrangement of n objects in a specific order
  using r objects at a time is called a permutation
  of nPr
• The formula is nPr
• 5P2
  20

5! (5 2)!
5! 5 4 3 2 1 2! 2
1
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Combinations

• Combinations are similar to permutationsbut are
  used when the order or sequence of the
  arrangement does not matter.
• The number of combinations or r objects selected
  from n objects is nCr
• nCr
• 4C2 6

n! (n r)! r!
4! 4! 4 3 2
1 (4-2)!2! 2!2! 2 1 2 1