Counting Theorem and Probability

8/6/2019 Counting Theorem and Probability

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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.


3/17

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

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Counting Theorem and Probability