bba 122 notes on probability.ppt

7/25/2019 BBA 122 notes on Probability.ppt

1/63

02/13/16 1


2/63

At the end of this topic you will be able to: Define probability Describe the classical, empirical and

subjective approaches to probability Understand the terms experiment, event,

outcome Define the terms conditional probability and

joint probability

02/13/16 2


3/63

Calculate probabilities using the rules of

addition and the rules of multiplication

02/13/16 3


4/63

Descriptive statistics is concerned with

summarizing data collected from past events !e turn to the second facet of statistics,

namely, computing the chance that somethingwill occur in the future

"his facet of statistics is called statistical

inferenceor inferential statistics

02/13/16 4


5/63

#tatistical inference deals with conclusions

about a population based on a sample ta$en

from that population

%ecause there is uncertainty in decisionma$ing, it is important that all the $nown ris$s

involved be scientifically evaluated

&elpful in this evaluation is probability theory,which has often been referred to as the science

of uncertainty

02/13/16 5


6/63

"he use of probability theory allows the

decision ma$er with only limited information

to analyze the ris$s and minimize the gamble

inherent, for example, in mar$eting a newproduct or accepting an incoming shipment

possibly containing defective parts

02/13/16 6


7/63

't is a number that describes the chance that

something will happen "his number lies

between zero and one

"he closer a probability is to (, the moreimprobable it is the event will happen

"he closer the probability is to ), the more sure

we are it will happen "hree $ey words areused in the study of probability: experiment,

outcomeand event

02/13/16 7


8/63

Experiment: is a process that leads to the

occurrence of one and only one of several

possible observations

An experiment has two or more possibleresults and it is uncertain which will occur for

example rolling a die

Outcome: this is a particular result of anexperiment

02/13/16 8


9/63

*or example, the tossing of a coin is an

experiment +ou may observe the toss of a

coin but you are unsure whether it will come

up heads- or tails- !hen one or more of the experiment-s

outcomes are observed, we call this an event

*or example all possible outcomes from a tossof a die are .), /, 0, 1, 2, 34

02/13/16 9


10/63

Event:this is a collection of one or more of an

experiment for example some possible events

from a roll of die are5 observe an even number,

observe a number greater than 0 or observe anumber 1 or less

02/13/16 10


11/63

Example Consider the following experiment "oss a coin

and observe whether the upside of the coin is

&ead or "ail "wo events may be occurred: &: &ead is observed,

": "ail is observed "he probability of an event A, denoted by 67A8,

in general, is the chance A will happen

02/13/16 11


12/63

"wo approaches to assigning probabilities are

the objectiveand subjectiveview points 9bjective probability is subdivided into ) Classical probability / mpirical probability

02/13/16 12


13/63

Classical probability

"his is based on the assumption that the

outcomes of an experiment are e;ually li$ely Using the classical view point, the probability

of an event happening is computed by dividing

the number of favorable outcomes by the

number of possible outcomes:

02/13/16 13


14/63

"he probability 67A8 of an event A is e;ual to

the number of possible simple events

7outcomes8 favorable to A divided by the total

number of possible simple events of theexperiment, ie,

02/13/16 14


15/63

02/13/16 15

where m< number of the simple

events into which the event A can be

decomposed


16/63

Example: Consider an experiment of rolling a six=sided

die !hat is the probability of the event >an even

number of spots face up #olution: "he possible outcomes are .), /, 0, 1, 2, 34

"he favorable outcomes ./, 1, 34 "herefore the probability of an even number an even

number@ and >an odd number@ in the die=tossing

experiment, then the set of events is collectively

exhaustive

*or the die tossing experiment, every outcome will be

either even or odd

"hus, in collective exhaustive events, at least one ofthe events must occur when an experiment is

conducted

02/13/16 19


20/63

Empirical probability:

Another way to define probability is based on

relative fre;uencies "he probability of an

event happening is determined by observingwhat fraction of the time similar events

happened in the past

02/13/16 20


21/63

'n terms of a formula: 6robability of event happening< umber of

times event occurred in the past? "otal number

of observations

02/13/16 21


22/63

Subjective probability

'f there is little or no past experience or

information on which to base a probability, it

may be arrived at subjectively ssentially, this means an individual evaluates

the available opinions and other information

and then estimates or assigns the probability

02/13/16 22


23/63

"his probability is aptly called a subjectiveprobability "hus, subjective concept of

probability means that a particular event

happening which is assigned by an individualbased on information available eg estimating

the li$elihood that you will earn an A in this

module

02/13/16 23


24/63

*or every event A of the field #, 'f the event A is decomposed into the mutually

exclusive events % and C belonging to # then

67A8


25/63

"he probability of any event A lies between (

and ):

02/13/16 25

( ) )( AP


26/63

Rules of addition

Special rule of addition "o apply the special rule of addition, the

events mutually exclusive 'f two events A and % are mutually exclusive,

the special rule of addition states that the

probability of one or the other event-soccurring e;uals the sum of their probabilities

02/13/16 26


27/63

ie 67A or %8< 67A8 B 67%8 *or three mutually exclusive events designated

A, % and C, the rule is written as 67A or % or C8< 67A8 B 67%8 B 67C8

02/13/16 27


28/63

iven the following information in the table

below:

Weight Event Number

ofpackages

Probabilit

y ofoccurrence

Underweight

A 100 0.025

Satisfactory B 3600 0.900Overweight C 300 0.075

ota! "000 1.000

02/13/16 28


29/63

!hat is the probability that a particular

pac$age will be either underweight or

overweightE

Solution 67A8 B 67C8< ((/2 B ((F2 < ()(

02/13/16 29


30/63

A Genn diagram is a useful tool to depict

addition or multiplication rules Assuming you have 0 mutually exclusive

events A, % and C, they can be illustrateddiagrammatically by use of Genn diagrams as

shown below:

02/13/16 30


31/63

#ventA

#ventB

#ventC

02/13/16 31


32/63

"his is used to determine the probability of an

event occurring by subtracting the probability

of the event not occurring from ) ie

( ) ( )APAP =)

02/13/16 32


33/63

A Genn diagram illustrating the complement

rule is shown as

#ventA A

02/13/16 33


34/63

Use the complementary rule to show the

probability of a satisfactory bag is (H(( #how the solution using a Genn diagram

Solution "he probability that the bag is unsatisfactory

e;uals:

67A8 B 67C8< ((/2 B ((F2


35/63

%ut the bag is satisfactory if it is not under or

overweight ie 67%8< )= .67A8 B 67C84

< )= ()(( < (H((

02/13/16 35


36/63

( ) H(((=CorA

A

0.025

C

0.075

02/13/16 36


37/63

A sample of employees of !orldwide nterprise is to

be surveyed about a new pension plan "he

employees are classified as follows5

Classication Event Number ofemployee

S%&ervisors A 120

'aintenance B 50

(rod%ction C 1"60

'anage)ent * 302

Secretaria! # 6+

02/13/16 37


38/63

a8 !hat is the probability that the first person

selected is: i8 either in maintenance or a secretaryE

ii8 not in managementEb8 Draw a Genn diagram illustrating your

answers to part a8

c8 Are the events in part a8 i8 complementaryor mutually exclusive or bothE

02/13/16 38


39/63

"he outcomes of an experiment may not be

mutually exclusive !hen two or more events both occur

concurrently, the probability is calledjointprobability

"he following Genn diagram shows three

events are not mutually exclusive

02/13/16 39


40/63

02/13/16 40


41/63

"his rule for two events designated A and % iswritten

67A or %8< 67A8 B 67%8= 67A and %8

IIIII) ;uation ) is what we call the general rule for

addition

02/13/16 41


42/63

"hus, if we compare the general and specificrules of addition, the important difference is

determining if the events are mutually

exclusive 'f the events are mutually exclusive, then the

joint probability 67A and %8< ( and we could

use the special rule of addition

02/13/16 42


43/63

)8 "he events A and % are mutually exclusive#uppose 67A8


44/63

Special rule of multiplication "his re;uires that two events A and % are

independent

"wo events are independent if the occurrenceof one event does not affect the probability of

the occurrence of the other event

02/13/16 44


45/63

*or two independent events A and %, theprobability that A and % will both occur is

found by multiplying the two probabilities ie

67A and %8< 67A867%8 *or 0 independent events A, % and C will be 67A and % and C8< 67A867%867C8

02/13/16 45


46/63

A survey by the American AutomobileAssociation 7AAA8 revealed 3(J of its

members made airline reservations last year

"wo members are selected at random !hat isthe probability both made airline reservations

last yearE

02/13/16 46


47/63

J3(87

J3(87

/

)

=

=

RP

RP

02/13/16 47


48/63

!here *irst and second members made reservation

#ince the number of AAA members is verylarge, you may assume that the above

members are independent "herefore5

/)andRR

02/13/16 48


49/63

( ) ( ) 03(3((3((87 /)/) === RPRPandRRP

02/13/16 49


50/63

"his is the probability of a particular eventoccurring, given that another event has

occurred

't can be written as Conditional probability of A given %

02/13/16 50


51/63

( ) ( )( )BP

AandBP

BAP =

02/13/16 51


52/63

Conditional probability of % given A

( ) ( )( )AP

AandBP

ABP =

02/13/16 52


53/63

#ymbolically, the joint probability 67A and %8 is foundby

( ) ( ) ( )( ) ( ) ( )

ABPAPAandBP

BAPBPAandBP

=

=

02/13/16 53


54/63

#uppose 67A8< (1( and 67%?A8


55/63

'f the number of possible outcomes in anexperiment is small, it is relatively easy to

count them

"here are six possible outcomes, for example,resulting from the roll of a die

02/13/16 55


56/63

'f there are m ways of doing one thing and nways of doing another thing, there are

ways of doing both

'n terms of a formula "otal number of arrangements < 7m87n8 "he multiplication formula is applied to find

the number of possible arrangements for / ormore groups

nm

02/13/16 56


57/63

's any arrangement of r objects selected from a

single group of n possible objectsote that the arrangements a b c and b a c are

different permutations "he formula to count the total number of

different permutations is

( )KK

rn

nP

rn

=

02/13/16 57


58/63

!here n< the total number of objects r< the number of objects selected nK means n7n=)87n=/87n=08IIIII)

*or instance 3K 37287187087/87)8


59/63

0 electronic parts are to be assembled into aplug=in unit for a "G set "he parts can be

assembled in any order 'n how many different

ways can the 0 parts be assembledE

( ) 3

)

)/0

K(

K0

K00

K0=

==

=rn

P

02/13/16 59


60/63

"he %etts Lachine shop inc, has eight screwmachines but only three spaces available in the

production area for the machines 'n how

many different ways can the eight machines bearranged in the three spaces availableE

02/13/16 60


61/63

'f the order of the selected objects is notimportant, any selection is called a

combination. "he formula to count the

number of r object combinations from a set ofn objects is5

( )KK

K

rnr

nC

rn

=

02/13/16 61


62/63

#olve the following

( )

)(

8)/087)/7

)/012

K/2K/

K2/2 =

=

=C

02/13/16 62


63/63

#olve the following

13

/F

0H

1F

C

C

P

P

bba 122 notes on probability.ppt

Documents