Post on 08-Apr-2018
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COUNTING THEOREM
Sample Space sum of outcome in any statistical experiment.S = _H, TaS = _1, 2 , 3, 4, 5, 6aS =_HH, HT, TH, TTa
Event is a subset of a sample space S.
Simple and Compound Events
If an event is a set containing only one element of a sample space, itis called a simple event.A compound event is one that can be expressed as the union ofsimple events.
Ex. Drawing a heart from a deck of 52 playing cards is a subset A =_ka of the sample space S = _ijklaTherefore A is a simple event.Event B of drawing a red card is a compound event since B = _k,j aNull Space is a subset of the sample space that contains noelements. We denoted the events by the symbol.
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COUNTING THEOREM
VENNDIAGRAM pictorial representation showing therelationships of events.
SET OPERATIONS
1. Intersection of events common to A and BEx. Let A = _1,2,3,4,5a and B = _2,4,6,8a then A B = _2,4a
2. Union of events either A or B or bothEx. Let A = _2,3,5,8a and B = _2,4,6,8a then A B = _2,3,5,6,8a
3. Mutually exclusive events A and B have no elements incommonEx. Let A = _2,4,6a and B = _1,3,5a then A B = .
4. Compliments of and event not in AEx. Let S = _dog, book, coin, map, cigarette, wara and A = _dog,book, cigarette, warathen A = _coin , mapa.
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COUNTING THEOREM
The Fundamental principles of counting oftenreferred to as a multiplication rule.
Theorem 1: Multiplication Rule
If an operation can be performed in n1 ways and iffor each of these, a second operation can beperformed in n2 ways, then the two operations canbe performed in n1*n2 ways.
Example: How many sample points are there in asample space if a pair of dice are thrown once?
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COUNTING THEOREM
Theorem 2: Generalized Multiplication RuleIf an operation can be performed in n1 ways and iffor each of these, a second operation can beperformed in n2 ways, if for each of the first two, a
third operation can be performed in n3 ways and soon, then the sequence of k operation can beperformed in n1*n2*n3*nk ways.
Example: How many lunch are possible consistingof soups, sandwiches, desserts and drinks if onehas to select from 4 soups, 3 sandwiches, 5desserts and 4 drinks?
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COUNTING THEOREM
PERMUTATION-an ordered arrangement of all orpart of set of object.*The number of permutation of n distinct objecttaken all together is n!
*The number of permutation of n distinct objecttaken r at a time is nPr.*The number of permutation of n distinct objectarranged in a circle in (n-1)!*The number of distinct permutations of n things of
which n1 are of one kind, n2 of a second kind.nk ofkth kind is
!!.....!
!
21 knnn
n
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COUNTING THEOREM
*The number of ways partitioning a set of objectsinto r cells with n1elements in the 1
stcell, n2elements in the 2nd cell and so on is
!!...!
!
... 2121 rr nnn
n
nnn
n!
where n1+ n2+ ..+ nr=n
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COUNTING THEOREM
EXAMPLES:1. How many distinct permutations can be made from the letters of
the word COLUMNS? INFINITY?2. How many of these permutations starts with the letter m?3. How many ways can the 5 starting position on a baseball team be
filled with 8 men who can play any of the position?
4. How many ways can 5 different trees can be planted in a circle?5. A college plays 12 football games during a season. In how many
ways can the team end the season with 7 wins, 3 loses and 2ties?
6. How many number consisting of five different digits, each can bemade from the digit1 to 9 if:
a) repetitions are allowedb) repetitions are not allowedc) numbers must be oddd) numbers must be evene) numbers must be less than 40,000f) the first two digits of the numbers are even
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COUNTING THEOREM
COMBINATIONS A set of things withoutreference to the order in which they arearranged.
*The number of combinations of distinct objectstaken r at a time is
!)!(
!
rrn
nC
rn
!
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COUNTING THEOREM
EXAMPLES:
1. In how many ways can 5 switches be chosen from a lot
containing15 good switches and 5 defectives switches?
2. If the 5 switches must be 3 good and 2 defectives.
3. From a group of 4 men and 5 women, how many committees
of size 3 are possiblea) with no restrictions
b) with 1 man and 2 women
c) with 2 men and1 woman if a certain must be on the
committee
4. A shipment of12 tv sets, contains 3 defective sets.In how many
ways can a hotel purchase 5 of that sets andreceive at least 2
of the defective sets?
PRA T ICESET 2
QUIZ2
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PROBABILITY
The probability of an event E that can occur in mways but of n equally likely outcomes is given by:
where m are the elements of the events and n arethe elements in the sample space.Theorems1: If an event does not occur then P (E) = 0.2. If an event E is contain to occur and every trialis a success, then P(E) = 1.3. The probability of an event E will occur is anumber from 0 to 1 only.
n
E !! )(
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PROBABILITY
Example: Suppose we draw a card froman ordinarydeck
ofcards. Howmanyequallylikelyoutcomesare there? What isthe probabilityofdrawingaqueen? What isthe probabilityofdrawingaheart? What isthe probabilityofdrawingaclub? What isthe probabilityofdrawingfour?
What isthe probabilityofdrawingared? What isthe probabilityofdrawingablack? What isthe probabilityofdrawingakingor
nine?
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PROBABILITY
ADDITIVE RULE
1. If A and B are mutually exclusive events then P (A
B) = P (A) + P (B)
2. If A and B are any two events then P (A B) = P (A)+ P (B) P (A B)3. If A1, A2, .An are any mutually exclusive eventsthen P (A1 A2 .. An) = P (A1) + P (A2) +P(An)
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PROBABILITY
Example1. Whatisthe probabilityofgetting7or11whenapairofdice istossed?
Example 2. The probabilitythatMae passedMathis2/3andthe probabilitythatshe passedEnglishis4/9. Ifthe
probabilityofpassingbothcoursesis, whatistheprobabilitythatshe passesone ofthese courses?
Example 3. The probabilityofanAmericanindustrywilllocate inmanilais0.7andthe probabilitythatitwilllocate inBrusselsis0.9andthe probabilitythatitwilllocate eitherManilaorBrusselsor bothis0.8. Whatisthe probabilitythatitwilllocate:a) inbothcities?b)Inneithercities?
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PROBABILITY
CONDITIONAL PROBABILITY
Example: Inanexperimenttostudythe dependence ofhypertensiononsmokinghabits,the followingdatawascollectedon180individuals.
Ifone individualisselectedatrandom,findthe probabilitythatthepersonis;
a)experiencinghypertension,giventhatapersonisaheavysmoker
b)non-smokergiventhatthe personisexperiencingnohypertension
)(
)()/(
AP
BAPABP
!
192648No hypertension
303621Hypertension
Heavy SmokersModerate SmokersNon-Smokers
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PROBABILITY
MULTIPLICATIVE RULE
P (A B) = P (A) * P (B/A)P (A1 A2 ..An) = P (A1)* P (A2/A1) * P (A3/A1A2)
Example: The probabilitythatamarriedmanwatchestheshowis0.4 andthe probabilitythatamarriedwomanwatchesthe showis0.5. The probabilitythatamanwatchesthe showgiventhatawife doesis0.7. Findthe probabilitythat:
a) amarriedcouple watchesthe showb)awife watchesthe showgiventhatherhusbanddoesc) atleastone personofamarriedcouple willwatchtheshow
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PROBABILITY
INDEPENDENT EVENTS
P (A B) = P (A) * P (B/A)P (A1 A2 An) = P (A1) * P (A2) * ..P (An)
Example: The probabilitythatapersonrecoversfromadelicate heartoperationis0.8. Whatisthe probabilitythat:a) 2ofthe next3patientwhohave thisoperationsurviveb)allofthe nextthree patientwhohave thisoperationsurvive
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PROBABILITY
PROBABILITY TREE
Example1:Medicine brandXhasan80%ofbeingeffectiveforpeople withbloodtype Oandonly50%chance ofbeingeffective forpeople withbloodtype A. MedicinebrandYhas90%ofbeingeffective forpeople withbloodtype Oand40%chance ofbeingeffective forpeople withbloodtype A. Amongthe populationonly70%are bloodtype Oandthe resthasbloodtype A. Whichmedicine hasagreatereffectivity?
Example2: Inacertainschool, ofthe studentsareleadersandthe remainingare theirfollowers. Itisknownthatthe probabilitythataleaderwillcooperate is1/3andthe probabilitythatafollowerwillcooperate is2/5. Whatisthe probabilitythatastudentwillcooperate?
PRACTICESET3 QUIZ3