Sample Space, S

1
Sample Space, S

• The set of all possible outcomes of an
  experiment.
• Each outcome is an element or member or sample
  point.
• If the set is finite (e.g., H/T on coin toss,
  number on the die, etc.)
• S H, T
• S 1, 2, 3, 4, 5, 6
• in general, S e1, e2, e3, , en
• where ei the outcomes of interest
• Note sometimes a tree diagram is helpful in
  determining the sample space

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Sample Space

• Example The sample space of gender and
  specialization of all BSE students in the School
  of Engineering is

3
Events

• A subset of the sample space reflecting the
  specific occurrences of interest.
• Example,
• All EVE students,
• V

4
Events

• Complement of an event, (A, if A is the event)
• e.g., students who are not EVE,
• Intersection of two events, (A n B)
• e.g., engineering students who are EVE and
  female,
• Mutually exclusive or disjoint events
• Union of two events, (A U B)

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Venn Diagrams

• Example, events V (EVE students) and F (female
  students)

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Other Venn Diagram Examples

• Mutually exclusive events
• Subsets

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Example

• Students who are male, students who are ECE,
  students who are on the ME track in ECE, and
  female students who are required to take ISE 412
  to graduate.

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Sample Points

• Multiplication Rule
• If event A can occur n1 ways and event B can
  occur n2 ways, then an event C that includes
  both A and B can occur
• n1 n2
• ways.
• Example, if there are 6 ways to choose a female
  engineering student at random and there are 6
  ways to choose a male student at random, then
  there are
• 6 6 36
• ways to choose a female and a male engineering
  student at random.

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Another Example

• Example 2.14, pg. 32

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Permutations

• definition an arrangement of all or part of a
  set of objects.
• The total number of permutations of the 6
  engineering specializations in MUSE is
• In general, the number of permutations of n
  objects is
• n!

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Permutations

• If we take the number of specializations 3 at a
  time (n 6, r 3), the number of permutations
  is
• In general,

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Example

• A new group, the MUSE Ambassadors, is being
  formed and will consist of two students (1 male
  and 1 female) from each of the BSE
  specializations. If a prospective student comes
  to campus, he or she will be assigned one
  Ambassador at random as a guide. If three
  prospective students are coming to campus on one
  day, how many possible selections of Ambassador
  are there?

13
Combinations

• Selections of subsets without regard to order.
• Example How many ways can we select 3 guides
  from the 12 Ambassadors?

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Probability

• The probability of an event, A is the likelihood
  of that event given the entire sample space of
  possible events.
• 0 P(A) 1 P(ø) 0 P(S) 1
• For mutually exclusive events,
• P(A1 U A2 U U Ak) P(A1) P(A2) P(Ak)