Statistical Experiments

JMB Chapter 2 Lecture 1 v3 EGR 252 Spring 2012 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


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


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… }


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}


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}


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


Other Venn Diagram Examples Five non-mutually exclusive events

Subset – The green circle is a subset of the beige circle


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


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 .


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


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


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


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


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)


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


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


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


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

Statistical Experiments

Documents

Transcript of Statistical Experiments