math30-6 lecture 5

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MATH30-6

Probability and Statistics

Preliminary Concepts on

Probability

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Objectives

At the end of the lesson, the students are expected to

Understand and describe sample spaces and events for

random experiments with graphs, tables, lists, or tree

diagrams; Use permutation and combinations to count the

number of outcomes in both an event and the sample

space;

Define probability; and Relate counting techniques to real life situations.

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Probability

A tool to relate the descriptive statistics to inferential

statistics

Ratio of number of samples derived from the total

population Deals with counting elements

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Random Experiment

An experiment that can result in different outcomes,even though it is repeated in the same manner every

time

Examples:

- Measuring a current in a copper wire with the presence

of uncontrollable inputs resulting the variations in

measurements

- Designing a communication system (computer or voicecommunication network) where the information

capacity available to serve individuals using the

network is an important design consideration

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Sample Space

The set of all possible outcomes of a randomexperiment

Denoted as S

Examples:

Consider the experiment of tossing a die.

Sample space for the number appearing on the top

face:

S1= {1, 2, 3, 4, 5, 6}

Sample space for the number appearing on the top

face whether it is even or odd:

S2= {even, odd}

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Discrete Sample Space

A sample space is discrete if it consists of a finite orcountable infinite set of outcomes.

Examples:

- Sample space for the number appearing on the top

face:

S= {1, 2, 3, 4, 5, 6}

- Sample space for a thrown die until a five occurs:

S= {F, NF, NNF, NNNF, }

where F= occurrence of 5 and N= nonoccurrence of 5.

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Continuous Sample Space

A sample space is continuous if it contains an interval(either finite or infinite) of real numbers.

Example:

- Sample space of the life in years (t) of a certain

electronic component:

S= {t|t 0}

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Sample Space

Provide a reasonable description of the sample space foreach of the random experiments in Exercises 2-1 to 2-17.

There can be more than one acceptable interpretation of

each experiment. Describe any assumptions you make.

2-2/28 Each of the four transmitted bits is classified as

either in error or not in error.

2-4/28 The number of hits (views) is recorded at a high-volume Web site in a day.

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Sample Space

Provide a reasonable description of the sample space foreach of the random experiments in Exercises 2-1 to 2-17.

There can be more than one acceptable interpretation of

each experiment. Describe any assumptions you make.

2-16/28 An order for a computer system can specify

memory of 4, 8, or 12 gigabytes, and disk storage of 200,

300, or 400 gigabytes. Describe the set of possible orders.

2-17/28 Calls are repeatedly placed to a busy phone line

until a connection is achieved.

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Sample Space

2-18/28 In a magnetic storage device, three attempts aremade to read data before an error recovery procedure

that repositions the magnetic head is used. The error

recovery procedure attempts three repositionings before

an abortmessage is sent to the operator. Let

sdenote the success of a read operation

fdenote the failure of a read operation

Fdenote the failure of an error recovery procedure

Sdenote the success of an error recovery procedureAdenote an abort message sent to the operator.

Describe the sample space of this experiment with a tree

diagram.

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Event

Subset of the sample space of a random experiment

Consider the events E1and E2.

Union of two events

- Consists of all outcomes that are contained in either of

the two events

- Denoted by E1E2

Intersection of two events

- Consists of all outcomes that are contained in both of

the two events

- Denoted by E1E2

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Event

Complement of an event

- Set of outcomes in the sample space that are not in the

event

- The complement of the event Eis Eor EC.

- (E)= E

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Event

2-19/28 Three events are shown on the Venn diagram inthe following figure:

Reproduce the figure and shade the region that

corresponds to each of the following events:

(a)A (b)AB (c) (AB) C

(d) (BC) (e) (AB)C

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Venn Diagram

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Venn Diagram

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Venn Diagram

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Event

2-21/28 A digital scale is used that provides weights to thenearest gram.

(a) What is the sample space for this experiment?

Let A denote the event that a weight exceeds 11 grams,

let Bdenote the event that weight is less than or equal to15 grams, and let C denote the event that a weight is

greater than or equal to 8 grams and less than 12 grams.

Describe the following events.

(b)AB (c)AB (d)A(e)ABC (f) (AC) (g)ABC

(h) BC (i)A(BC)

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Event

2-23/29 Five bits are transmitted over a digitalcommunications channel. Each bit is either distorted or

received without distortion. Let Aidenote the event that

the ith bit is distorted, i= 1, , 5.

(a) Describe the sample space for this experiment.(b) Are theAismutually exclusive?

Describe the outcomes in each of the following events:

(c)A1

(d)A1

(e)A1A2A3A4

(f) (A1A2) (A3A4)

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Counting Techniques

An important part of combinatorics (study ofarrangement of objects which is part of discrete

mathematics)

Methods used for counts of the numbers of outcomes

in the sample space and various events for analyzingrandom experiments

Used for more complicated problems and more difficult

sample space or an event

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Multiplication Rule

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Generalized Multiplication Rule

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Examples:

2-16/46 How many even four-digit numbers can be

formed from the digits 0, 1, 2, 5, 6, and 9 if each digit can

be used only once?

2-30/51 In how many different ways can a true-false test

consisting of 9 questions can be answered?

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2-34/34 A wireless garage door opener has a codedetermined by the up or down setting of 10 switches.

How many outcomes are in the sample space of possible

codes?

2-35/35 An order for a computer can specify any one of

five memory sizes, any one of three types of displays, and

any one of five sizes of hard disks, and can either include

or not include a pen tablet? How many different systemscan be ordered?

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14/340 In a version of the computer language BASIC, thename of a variable is a string of one or two alphanumeric

characters, where uppercase and lowercase letters are

not distinguished. (An alphanumeric character is either

one of the 26 English letters or one of the 10 digits.)Moreover, a variable name must begin with a letter and

must be different from the five strings of two characters

that are reserved for programming use. How many

different variable names are there in this version of

BASIC?

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15/340 Each user on a computer system has a password,which is six to eight characters long, where each character

is an uppercase letter or a digit. Each password must

contain at least one digit. How many possible passwords

are there?

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Permutation

An ordered sequence of the elementsExample:

Consider the three letters a, b, and c. The possible

permutations are abc, acb, bac, bca, cab, and cba.

The number of permutations of ndifferent elements is

n! (read as nfactorial) where

n! = n(n 1) (n 2) 2 1

Note: 0! = 1

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Permutation of Subsets

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Permutation

Examples:2-42/30 In the layout of a printed circuit board (PCB) for

an electronic product, there are 15 different locations that

can accommodate chips.

(a) If five different types of chips are to be placed on theboard, how many different layouts are possible?

(b) If the five chips that are placed on the board are of the

same type, how many different layouts are possible?

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Permutation

2-45/31 Consider the design of a communication system.(a) How many three-digit phone prefixes that are used to

represent a particular geographic area (such as an area

code) can be created from the digits 0 through 9?

(b) As in part (a), how many three-digit phone prefixes arepossible that do not start with 0 or 1, but contain 0 or 1 as

the middle digit?

(c) How many three-digit phone prefixes are possible in

which no digit appears more than once in each prefix?

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Permutation

2.31/51 A witness to a hit-and-run accident told the policethat the license number contained the letters RLHfollowed by 3 digits, the first of which was a 5. If thewitness cannot recall the last 2 digits, but is certain thatall 3 digits are different, find the maximum number of

automobile registrations that the police may have tocheck.

2.32/52 (a) In how many ways can 6 people be lined up to

get on a bus?(b) If 3 specific persons, among 6, insist on following eachother, how many ways are possible?

(c) If 2 specific persons, among 6, refuse to follow eachother, how many ways are possible?

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Permutation

2.33/52 If a multiple-choice test consists of 5 questionseach with 4 possible answers of which only 1 is correct,

(a) In how many different ways can a student check offone answer to each question?

(b) In how many ways can a student check off one answerto each question and get all the answers wrong?

2.36/52 (a) How many three-digit numbers can be formedfrom the digits 0, 1, 2, 3, 4, 5, and 6 if each digit can be

used only once?(b) How many of these are odd numbers?

(c) How many are greater than 330?

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Circular Permutation

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Permutation of Similar Objects

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Permutation of Similar Objects

Examples:2-45/52 How many distinct permutations can be made

from the letters of the word INFINITY?

2-46/52 In how many ways can 3 oaks, 4 pines, and 2maples be arranged along a property line if one does not

distinguish among trees of the same kind?

2-12/26 A part is labeled by printing with four thick lines,three medium lines, and two thin lines. If each ordering of

the nine lines represent a different label, how many

different labels can be generated by using this scheme?

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Ordered Partition

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Combination

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Combination

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Combination

Examples:2-14/27 Sampling without Replacement

A bin of 50 manufactured parts contains three defective

parts and 47 nondefective parts. A sample of six parts is

selected from the 50 parts without replacement. That is,each part can only be selected once and the sample is a

subset of the 50 parts. How many different samples are

there of size six that contain exactly two defective parts?

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Combination

2-22/50 A young boy asks his mother to get five game-boycartridges from his collection of 10 arcade and 5 sports

games. How many ways are there that his mother will get

3 arcade and 2 sports games, respectively?

11/358 How many poker hands of five cards can be dealt

from a standard deck of 52 cards? Also, how many ways

are there to select 47 cards from a standard deck of 52

cards?

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Combination

12/360 How many ways are there to select five playersfrom a 10-member tennis team to make a trip to a match

at another school?

13/360 A group of 30 people have been trained asastronauts to go on the first mission to Mars. How many

ways are there to select a crew of six people to go on this

mission (assuming that all crew members have the same

job)?

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Combination

15/360 Suppose that there are 9 faculty members in themathematics department and 11 in the computer science

department. How many ways are there to select a

committee to develop a discrete mathematics course at a

school if the committee is to consist of three facultymembers from the mathematics department and four

from the computer science department?

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Summary

A random experiment is an experiment that can resultin different outcomes, even though it is repeated in the

same manner each time.

The sample space is the set of all possible outcomes of

a random experiment. The event is a subset of a sample space.

Multiplication rule is a formula used to determine the

number of ways to complete an operation from the

number of ways to complete successive steps. A permutation is an arrangement of all (n!) or part

(permutation of subsets) of a set of objects.

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Summary

Two circular permutations are not considered differentunless corresponding objects in the two arrangements

are preceded or followed by a different object as we

proceed in a clockwise direction. For example, if 4

people are playing bridge, we do not have a new

permutation if they all move one position in a

clockwise direction. By considering one person in a

fixed position and arranging the other in 3! ways, we

find that there are 6 distinct arrangements for the

bridge game.

A combination is actually a partition with two cells, the

on cell containing the robjects selected and the other

cell containing the (nr) objects that are left.

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References

Montgomery and Runger. Applied Statistics and

Probability for Engineers, 5thEd. 2011

Rosen, Kenneth H. Discrete Mathematics and Its

Applications, 6thEd. 2007

Walpole, et al. Probability and Statistics for Engineers

and Scientists 9thEd. 2012, 2007, 2002

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