Statistical Experiments

1
Statistical Experiments

• The set of all possible outcomes of an experiment
  is the Sample Space, S.
• Each outcome of the experiment is an element or
  member or sample point.
• If the set of outcomes is finite, the outcomes in
  the sample space can be listed as shown
• S H, T
• S 1, 2, 3, 4, 5, 6
• in general, S e1, e2, e3, , en
• where ei each outcome of interest

2
Tree Diagram

• If the set of outcomes is finite sometimes a tree
  diagram is helpful in determining the elements in
  the sample space.
• The tree diagram for students enrolled in the
  School of Engineering by gender and degree
• The sample space
• S MEGR, MIDM, MTCO, FEGR, FIDM, FTCO

3
Your Turn Sample Space

• Your turn The sample space of gender and
  specialization of all BSE students in the School
  of Engineering is
• or
• 2 genders, 6 specializations,
• 12 outcomes in the entire sample space

S FECE, MECE, FEVE, MEVE, FISE, MISE, FMAE,
etc
S BMEF, BMEM, CPEF, CPEM, ECEF, ECEM, ISEF,
ISEM
4
Definition of an Event

• A subset of the sample space reflecting the
  specific occurrences of interest.
• Example In the sample space of gender and
  specialization of all BSE students in the School
  of Engineering, the event F could be the student
  is female
• F BMEF, CPEF, ECEF, EVEF, ISEF, MAEF

5
Operations on Events

• Complement of an event, (A, if A is the event)
• If event F is students who are female,
• F BMEM, EVEM, CPEM, ECEM, ISEM, MAEM
• Intersection of two events, (A n B)
• If E environmental engineering students and F
  female students,
• (E n F) EVEF
• Union of two events, (A U B)
• If E environmental engineering students and I
  industrial engineering students,

(E U I) EVEF, EVEM, ISEF, ISEM
6
Venn Diagrams

• Mutually exclusive or disjoint events
• Male Female
• Intersection of two events
• Let Event E be EVE students (green circle)
• Let Event F be female students (red circle)
• E n F is the overlap brown area

7
Other Venn Diagram Examples

• Five non-mutually exclusive events
• Subset The green circle is a subset of the
  beige circle

8
Subset Examples

• Students who are male
• Students who are ECE
• Students who are on the ME track in ECE
• Female students who are required to take ISE 428
  to graduate
• Female students in this room who are wearing
  jeans
• Printers in the engineering building that are
  available for student use

9
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 different female students
  and 6 different male students in the room, then
  there are
• 6 6 36 ways to
  choose a team consisting of a female and a male
  student .

10
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
• 654321 720
• In general, the number of permutations of n
  objects is n!

NOTE 1! 1 and 0! 1
11
Permutation Subsets

• In general,
• where n the total number of distinct items
  and r the number of items in the subset
• Given that there are 6 specializations, if we
  take the number of specializations 3 at a time (n
  6, r 3), the number of permutations is

12
Permutation 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?
• If the outcome is defined as ambassador assigned
  to student 1, ambassador assigned to student 2,
  ambassador assigned to student 3
• Outcomes are A1,A2,A3 or A2,A4,A12 or A2,
  A1,A3 etc
• Total number of outcomes is 12P3 12!/(12-3)!
  1320

13
Combinations

• Selections of subsets without regard to order.
• Example How many ways can we select 3 guides
  from the 12 Ambassadors?
• Outcomes are A1,A2,A3 or A2,A4,A12 or A12,
  A1,A3 but not A2,A1,A3
• Total number of outcomes is
• 12C3 12! / 3!(12-3)! 220

14
Introduction to Probability

• The probability of an event, A is the likelihood
  of that event given the entire sample space of
  possible events.
• P(A) target outcome / all possible outcomes
• 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)

15
Calculating Probabilities

• Examples
• There are 26 students enrolled in a section of
  EGR 252, 3 of whom are BME students. The
  probability of selecting a BME student at random
  off of the class roll is
• P(BME) 3/26 0.1154
• 2. The probability of drawing 1 heart from a
  standard 52-card deck is
• P(heart) 13/52 1/4

16
Additive Rules

• Experiment Draw one card at random from a
  standard 52 card deck. What is the probability
  that the card is a heart or a diamond?
• Note that hearts and diamonds are mutually
  exclusive.
• Your turn What is the probability that the card
  drawn at random is a heart or a face card
  (J,Q,K)?

17
Your Turn Solution

• Experiment Draw one card at random from a
  standard 52 card deck. What is the probability
  that the card drawn at random is a heart or a
  face card (J,Q,K)?
• Note that hearts and face cards are not mutually
  exclusive.

P(H U F) P(H) P(F) P(HnF) 13/52
12/52 3/52 22/52
18
Card-Playing Probability Example

• P(A) target outcome / all possible outcomes
• Suppose the experiment is being dealt 5 cards
  from a 52 card deck
• Suppose Event A is 3 kings and 2 jacks
• K J K J K K K K J J (combination or
  perm.?)
• P(A)
  9.23E-06

combinations(3 kings)
combinations(2 jacks)