jmb chapter 2 lecture 1 v3egr 252.001 2011slide 1 statistical experiments the set of all possible...
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JMB Chapter 2 Lecture 1 v3 EGR 252.001 2011 Slide 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
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JMB Chapter 2 Lecture 1 v3 EGR 252.001 2011 Slide 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}
S
M F
EGR IDM TCO EGR IDM TCO
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JMB Chapter 2 Lecture 1 v3 EGR 252.001 2011 Slide 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… }
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JMB Chapter 2 Lecture 1 v3 EGR 252.001 2011 Slide 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}
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JMB Chapter 2 Lecture 1 v3 EGR 252.001 2011 Slide 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 ∩ B) If E = environmental engineering students and F =
female students,
(E ∩ 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}
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JMB Chapter 2 Lecture 1 v3 EGR 252.001 2011 Slide 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 ∩ F is the overlap – brown area
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JMB Chapter 2 Lecture 1 v3 EGR 252.001 2011 Slide 7
Other Venn Diagram Examples Five non-mutually exclusive events
Subset – The green circle is a subset of the beige circle
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JMB Chapter 2 Lecture 1 v3 EGR 252.001 2011 Slide 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
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JMB Chapter 2 Lecture 1 v3 EGR 252.001 2011 Slide 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 .
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JMB Chapter 2 Lecture 1 v3 EGR 252.001 2011 Slide 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 …
6*5*4*3*2*1 = 720
In general, the number of permutations of n objects is n!
NOTE: 1! = 1 and 0! = 1
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JMB Chapter 2 Lecture 1 v3 EGR 252.001 2011 Slide 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
!!
rn
nPrn
4*5*6!36
!636
P
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JMB Chapter 2 Lecture 1 v3 EGR 252.001 2011 Slide 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
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JMB Chapter 2 Lecture 1 v3 EGR 252.001 2011 Slide 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
!!
!
rnr
n
r
n !!
!
rnr
nCrn
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JMB Chapter 2 Lecture 1 v3 EGR 252.001 2011 Slide 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)
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JMB Chapter 2 Lecture 1 v3 EGR 252.001 2011 Slide 15
Calculating Probabilities Examples:1. 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
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JMB Chapter 2 Lecture 1 v3 EGR 252.001 2011 Slide 16
Additive RulesExperiment: 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)?
5.052
26
52
13
52
13)()()( 2121 APAPAAP
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JMB Chapter 2 Lecture 1 v3 EGR 252.001 2011 Slide 17
Your Turn: SolutionExperiment: 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(H∩F)
= 13/52 + 12/52 – 3/52 = 22/52
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JMB Chapter 2 Lecture 1 v3 EGR 252.001 2011 Slide 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
3
4
5
52
0
44
2
4
3
4
combinations(3 kings) = combinations(2 jacks) =
2
4