LEAVING CERTIFICATE. ARRANGEMENTS, SELECTIONSANDPROBABILITY

This presentation will emphasise:The advantages of the counting methodThe importance of learning by doingClear thinking over blind application of rulesThe advantages of an experimental/investigative approachesThe suitability of PowerPoint as a teaching tool

Why Discrete Maths? Short syllabus Two Questions on Examination Paper (Possible third) Least popular questions attempted by students: Q.6 - 68% Q.7- 57%Important at Third Level Business - Science Psychology Medicine etc

Higher Syllabus

FUNDAMENTAL PRINCIPLE OF COUNTINGFundamental Principle of Counting: If one task can be accomplished in x different ways, and following this another task can be accomplished in y different ways, then the first task followed by the second can be accomplished in xy different ways.Example: A car manufacturer makes 3 models of cars, a mini, a saloon and an estate. These cars are all available in a choice of 5 colours; red, green, blue, black and orange. How many different cars are available ?35= 15 different versions

Example: A car manufacturer makes 3 models of cars, a mini, a saloon and an estate. These cars are all available in a choice of 5 colours; red, green, blue, black and orange. How many different cars are available ?15 different versionsA tree diagram can be used to illustrate this example

Example. In a restaurant you can have 6 types of muffin, 8 varieties of sandwich, and 5 drinks ( coffee, tea, coke, fanta, 7 up). Lunch is either a muffin and a hot drink or a sandwich and a cold drink. How many different choices of lunch are possible?

PERMUTATIONS (ARRANGEMENTS)In how many ways can we arrange the letters a,b,c,d,etaking the letters two at a time?Method 1.Write out all the arrangements(a,b) (a,c) (a,d) (a,e) (b,c) (b,d) (b,e) (c,d) (c,e) (d,e)Total = 20Method 2.Box Method54PossibilitiesTotal = 5x4 = 20Method 3.Use the P notation(b,a) (c,a) (d,a) (e,a) (c,b) (d,b) (e,b) (d,c) (e,c) (e,d)

NOTES ON THE P NOTATIONOn calculator6P2=Answer is 30SPECIAL ONES

Possibilities

Treat the vowels as one unit

ARRANGING THINGS IN A CIRCLEA circular permutation is not considered to be distinct from anotherunless one object is preceded or succeeded by a different object in both permutations

Are these different arrangements?NO. ALL FOUR ARE REPRESENTATIONS OF THE SAME ARRANGEMENT

COMBINATIONS (SELECTIONS)In how many ways can we select the letters a,b,c,d,e two at a time.Method 1.Write out all the selections(a,b) (a,c) (a,d) (a,e) (b,c) (b,d) (b,e) (c,d) (c,e) (d,e)NOTE : (b,a) is the same selection as (a,b)Total = 10Method 2.Use the C notationFormal Definition

We use this symbolOrigin of Formal Definition

NOTES ON THE C NOTATION

TWO IMPORTANT RULES

HEADS OR TAILS

Four distinguishable pairs of socks are in a drawer. If two socks are selected at random find the probability that a pair is chosen.Socks Problem

Fundamental Basis of ProbabilityNumber of possible outcomes is also called the Sample Space

Favourable Outcomes :Total Outcomes :A bag contains five blue and six red discs. A disc is drawn at random from the bag. Find the probability that the disc is red. From 6 red choose 1From 11 discs choose 1

Favourable Outcomes :Total Outcomes :A bag contains five blue and six red discs. Two discs are drawn at random from the bag. Find the probability that both discs are red. From 6 red choose 2From 11 discs choose 2

Favourable Outcomes :A bag contains five blue and six red discs. Two discs are drawn at random from the bag. Find the probability that exactly one red disc is drawn. Find the probability that one red and one blue disc are drawnFrom 6 red choose 1 and from 5 blue choose 1

Favourable Outcomes :Total Outcomes :A bag contains five blue and six red discs. Three discs are drawn at random from the bag. Find the probability that all three discs are red. From 6 red choose 3From 11 discs choose 3

Favourable Outcomes :A bag contains five blue and six red discs. Three discs are drawn at random from the bag. Find the probability that exactly two of the discs are red. From 6 red choose 2 and from 5 blue choose 1Note: Since 3 discs are drawn the third one must be considered and is blue

Unfavourable Outcomes :A bag contains five blue and six red discs. Three discs are drawn at random from the bag. Find the probability that at least one red disc is drawn. From 5 blue choose 3Note: This is the negative of drawing three blue discs.Total Outcomes :From 11 discs choose 3Favourable Outcomes :

Favourable Outcomes :A bag contains five blue and six red discs. Three discs are drawn at random from the bag. Find the probability that at least one red disc is drawn. From 6 red choose 3 Note: We can also do this by counting all of the cases that will yield a favourable outcome.orFrom 6 red choose 2 and from 5 blue choose 1orFrom 6 red choose 1 and from 5 blue choose 2Another method

RELATIVE FREQUENCY

Example 1.A drawing-pin can land point up or point down when dropped.Pat drops a drawing-pin 100 times and it lands point up35 times. Estimate the probability of the drawing pin landing point up.Solution:Probability =

Example 2A coin was tossed 1000 times and the number of heads were recorded. It was found that a head occurred 495 times. Estimate the probability of getting a head. Solution:Probability = Note: As the number of trials increases the relative frequency tends to stabilise about a constant value. This constant value is called the probability of the event occurring.Therefore the probability of getting a head is 0.5 or 50%

PROBABILITYThis can be useful in certain questions but is not required

SINGLE EVENT PROBLEMS

MULTIPLE EVENTS PROBLEMS. F.P.CMethod 1. Sample Space

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1,H

2,H

3,H

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6,H

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1,T

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MULTIPLE EVENT PROBLEMS.Ex. 1 An unbiased die is thrown twice. Find the probability of getting two equal scores or a total of 10

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1999 Q. 9 (a) An unbiased die is thrown twice. Find the probability of getting a total less than four.

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REPLACEMENT AND NO REPLACEMENTNote: Replacement does not occur unless explicitly stated

ORDER DOES NOT MATTER No. of favourable outcomes =

2005 Discrete maths questions

2005 Total Outcomes :From remaining cards choose 3.

From remaining cards choose 3

Only Two Favourable Outcomes

What the Chief Examiner had to say about Q6 -2005 (c) Parts (i) and (ii) involved relatively standard counting techniques in a familiar context. Nonetheless they required some clarity of thought, Part (iii) was poorly answered. Candidates seemed unable to identify favourable outcomes and many did not even attempt the question. It would appear that, other than with certain well-rehearsed question types, candidates lack the capacity to systematically list and/or systematically count outcomes satisfying particular criteria. Even the basic generic skills for this topic, such as exploring the situation by looking at examples of outcomes that satisfy or do not satisfy relevant criteria, were lacking.

2005

What the Chief Examiner had to say about Q7 2005(b) Parts (i) and (ii) were very well answered by most candidates. Parts (iii) and (iv), however, caused much more difficulty than ought to be expected at this level. It is worth noting that the majority of candidates approached these parts using multiplicative laws for probability, which are not on the core course. Such candidates (unless they have developed their understanding substantially through studying the material on the option) tend to favour a blind application of rules over clear thinking, and generally suffer the consequences in all but the most basic of situations. In this case, they made errors related to ordering. As might be expected, the candidates who stuck to the basic principle for dealing with all situations involving equally likely outcomes (i.e., count the outcomes and put favourable over possible)fared much better than those attempting to use the more complex rules.

LC 2003 Paper 2 Question 6 (a) (i)(a) Eight people, including Kieran and Anne, are available to form a committee.Five people must be chosen for the committee(i) In how many ways can a committee be formed if both Kieran and Anne must be on the chosen ? How do you get pupils to decide between combinations and permutations ?

LC 2003 Paper 2 Question 6 (a) (i)(a) Eight people, including Kieran and Anne, are available to form a committee.Five people must be chosen for the committee(i) In how many ways can a committee be formed if both Kieran and Anne must be on the chosen ? How do you get pupils to decide between combinations and permutations ?

LC 2003 Paper 2 Question 6 (a) (i)(a) Eight people, including Kieran and Anne, are available to form a committee.Five people must be chosen for the committee(i) In how many ways can a committee be formed if both Kieran and Anne must be on the chosen ? How do you get pupils to decide between combinations and permutations ?If combinations is being used what are the two important questions ? How many are available ? nHow many am I choosing/picking/selecting ? r

Committee (of 5)In how many ways can a committee be formed if both Kieran and Anne must be on the chosen ? ? ? ?How many are available ? nHow many am I choosing/picking/selecting ? r

Committee (of 5)KieranAnneHow many are available ? nHow many am I choosing/picking/selecting ? r? ? ? ? ?(ii) In how many ways can a committee be formed if neither Kieran nor Anne can be chosen

LC 2003 Paper 2 Question 6 (c)Ten discs, each marked with a different whole number from 1 to 10, are placed in a box. Three of the discs are drawn at random (without replacement) from the box.What is the probability that the disc with the number 7 is drawn ?What is the probability that the three numbers on the discs drawn are odd ?What is the probability that the product of the three numbers on the discs drawn is even ?What is the probability that the smallest number on the discs drawn is 4 ?

What is the probability that the disc with the number 7 is drawn ?Method F.P.CFavourable Outcomes :Total Outcomes :From the 10 discs choose any 3 A seven and two other discs

(ii) What is the probability that the three numbers on the discs are odd ?Method F.P.CFavourable Outcomes :Total Outcomes :From the 10 discs choose any 3 Three odd discs

Unfavourable Outcomes :Total Outcomes :From the 10 discs choose any 3 Three odd discs(iii) What is the probability that the product of the three numbers on the discs is even ?Note: If any or all the numbers are even then the product will be even. Favourable Outcomes = Total unfavourable =

(iv) What is the probability that the smallest number on the discs drawn is 4 ?Favourable Outcomes :Total Outcomes :From the 10 discs choose any 3 A four and two other discs larger than 4

LCH 2003 Paper 2 Question 7 (a) (i)Five cars enter a car park. There are exactly five vacant spaces in the car park.(i) In how many different ways can the five cars park in the vacant spaces ?543215.4.3.2.1 = 120 ways

LCH 2003 Paper 2 Question 7 (a) (ii)Two of the cars leave the car park without parking. In how many different ways can the remaining three cars park in the five vacant spaces ?There are 5 different spaces this car could choose. Similarly, only 3 spaces are left for the final car.No. of ways 5 . 4 . 3 = 60 However once a space is picked there is only 4 spaces left for the next car.

LCH 2003 Paper 2 Question 7 (b) (i)L and k are distinct parallel lines. a, b,c and d are points on L such that |ab|=|bc|=|cd|=1cm. x,y and z are points on K such that |xy|=|yz|=1cm. (i)How many different triangles can be constructed using three of the named points as vertices?Two identified methods here. Traditional C NotationInspection

LCH 2003 Paper 2 Question 7 (b) (i)L and k are distinct parallel lines. a, b,c and d are points on L such that |ab|=|bc|=|cd|=1cm. x,y and z are points on K such that |xy|=|yz|=1cm. (i)How many different triangles can be constructed using three of the named points as vertices.Two identified methods here. Traditional C NotationChoose 1 point from line L and two points from line KOR Choose 1 point from line K and two points from line L4.3 + 6.3 = 12+18 = 30 triangles

LCH 2003 Paper 2 Question 7 (b) (i)Land k are distinct parallel lines. A, b,c and d are points on L such that |ab|=|bc|=|cd|=1cm. x,y and z are points on K such that |xy|=|yz|=1cm. (i)How many different triangles can be constructed using three of the named points as vertices.Two identified methods here. (2) InspectionUsing just point a

LCH 2003 Paper 2 Question 7 (b) (i)Land k are distinct parallel lines. A, b,c and d are points on L such that |ab|=|bc|=|cd|=1cm. x,y and z are points on K such that |xy|=|yz|=1cm. (i)How many different triangles can be constructed using three of the named points as vertices.Two identified methods here. (2) InspectionUsing just point a2

LCH 2003 Paper 2 Question 7 (b) (i)Land k are distinct parallel lines. A, b,c and d are points on L such that |ab|=|bc|=|cd|=1cm. x,y and z are points on K such that |xy|=|yz|=1cm. (i)How many different triangles can be constructed using three of the named points as vertices.Two identified methods here. (2) InspectionUsing just point a33 triangles from a4 points (a,b,c,d)4 x 3 = 12 triangles

LCH 2003 Paper 2 Question 7 (b) (i)Land k are distinct parallel lines. A, b,c and d are points on L such that |ab|=|bc|=|cd|=1cm. x,y and z are points on K such that |xy|=|yz|=1cm. (i)How many different triangles can be constructed using three of the named points as vertices.Two identified methods here. (2) InspectionUsing just point x3

LCH 2003 Paper 2 Question 7 (b) (i)Land k are distinct parallel lines. A, b,c and d are points on L such that |ab|=|bc|=|cd|=1cm. x,y and z are points on K such that |xy|=|yz|=1cm. (i)How many different triangles can be constructed using three of the named points as vertices.Two identified methods here. (2) InspectionUsing just point x

LCH 2003 Paper 2 Question 7 (b) (i)Land k are distinct parallel lines. A, b,c and d are points on L such that |ab|=|bc|=|cd|=1cm. x,y and z are points on K such that |xy|=|yz|=1cm. (i)How many different triangles can be constructed using three of the named points as vertices.Two identified methods here. (2) InspectionUsing just point x

LCH 2003 Paper 2 Question 7 (b) (i)Land k are distinct parallel lines. A, b,c and d are points on L such that |ab|=|bc|=|cd|=1cm. x,y and z are points on K such that |xy|=|yz|=1cm. (i)How many different triangles can be constructed using three of the named points as vertices.Two identified methods here. (2) InspectionUsing just point x

LCH 2003 Paper 2 Question 7 (b) (i)Land k are distinct parallel lines. A, b,c and d are points on L such that |ab|=|bc|=|cd|=1cm. x,y and z are points on K such that |xy|=|yz|=1cm. (i)How many different triangles can be constructed using three of the named points as vertices.Two identified methods here. (2) InspectionUsing just point x

LCH 2003 Paper 2 Question 7 (b) (i)Land k are distinct parallel lines. A, b,c and d are points on L such that |ab|=|bc|=|cd|=1cm. x,y and z are points on K such that |xy|=|yz|=1cm. (i)How many different triangles can be constructed using three of the named points as vertices.Two identified methods here. (2) InspectionUsing just point x6 triangles from x3 points (x,y,z)3 x 6 = 18 triangles12 red + 18 blue = 30 triangles

LCH 2003 Paper 2 Question 7 (b) (ii)(ii) How many different quadrilaterals can be constructed ?Again two methods could be used here. Traditional C NotationInspection

LCH 2003 Paper 2 Question 7 (b) (ii)(ii) How many different quadrilaterals can be constructed ?Again two methods could be used here. Traditional C NotationQuadrilateral:Choosing 2 points from line L

Choosing 2 points from line K6 x 3 = 18 different quadrilaterals

LCH 2003 Paper 2 Question 7 (b) (ii)

(ii) How many different quadrilaterals can be constructed ?Again two methods could be used here. (2) InspectionUsing just [ab]

LCH 2003 Paper 2 Question 7 (b) (ii)

(ii) How many different quadrilaterals can be constructed ?Again two methods could be used here. (2) InspectionUsing just [ab]

LCH 2003 Paper 2 Question 7 (b) (ii)

(ii) How many different quadrilaterals can be constructed ?Again two methods could be used here. (2) InspectionUsing just [ab]There are just 3 quadrilaterals which can be formed using [ab]There are 6 possible line segments ([ab],[ac],[ad],[bc],[bd],[cd])6 x 3 = 18 different quadrilaterals

LCH 2003 Paper 2 Question 7 (b) (iii)

(iii)how many different parallelograms can be constructed using four of the named points as vertices ?[ab]

LCH 2003 Paper 2 Question 7 (b) (iii)

(iii)how many different parallelograms can be constructed using four of the named points as vertices ?[ab]

LCH 2003 Paper 2 Question 7 (b) (iii)

(iii)how many different parallelograms can be constructed using four of the named points as vertices ?3[ac]

LCH 2003 Paper 2 Question 7 (b) (iii)

(iii)how many different parallelograms can be constructed using four of the named points as vertices ?4[bc]

LCH 2003 Paper 2 Question 7 (b) (iii)

(iii)how many different parallelograms can be constructed using four of the named points as vertices ?5[bc]

LCH 2003 Paper 2 Question 7 (b) (iii)

(iii)how many different parallelograms can be constructed using four of the named points as vertices ?6[bd]

LCH 2003 Paper 2 Question 7 (b) (iii)

(iii)how many different parallelograms can be constructed using four of the named points as vertices ?7[cd]

LCH 2003 Paper 2 Question 7 (b) (iii)

(iii)how many different parallelograms can be constructed using four of the named points as vertices ?8[cd]8 parallelograms

LCH 2003 Paper 2 Question 7 (b) (iv)

(iv) If one quadrilateral is constructed at random, what is the probability that it is not a parallelogram ?Again, keep pupils on alert for total the opposite of what we are looking for.P(Success) = 1- P(Failure)Quadrilateral which is not a parallelogram = 18-8Parallelogram P( Quadrilateral which is not a parallelogram) =Quadrilateral which is not a parallelogram or

2004 Question 6 (c)253416

253416We want to pick 4 cards all different in colourFavourable Outcomes =Total Outcomes === 70

OOEE253416We want to pick 4 cards 2 odd and 2 evenColour does not matter here.Favourable Outcomes =From 4 odd pick 2 and from 4 even pick 2

253416OOEEExampleOOOOEEEEFrom the cards that are left we have one option: ie. to pick the even blue and the even yellow

(i)Possibilities = 8 7 6 5 4 3 2 1 = 8!

2004 Question 7The first runner can go in any one of 8 lanes. The second runner can now occupy any of the 7 remaining lanes.And so on until the eight runner has only one lane left

(ii)Possibilities2004 Question 7The first runner can go in any one of 8 lanes. The second runner can now occupy any of the 7 remaining lanes.And so on until the fifth runner has 4 lanes left = 8 7 6 5 4 =

These students do not study Biology = 21Total number of students = 56

This gives a total of 26 who studyat least two subjectsThese 4 do not study Biology

Total of 28 study PhysicsFavourable = Total =

2006 Question 6 (a)How many are available ? nHow many am I choosing/picking/selecting ? r

2006 Question 6 (c)

2006 Question 6 (c)

2006 Question 6 (c)orProbability = 1- P( all seven have the same birthday)

152006 Question 7 (a)

2006 Question 7 (b)(iv)P(matching at least three numbers)= P( match three or match four or match five)

Real LottoThe Odds Source www.lotto.ie Odds on winning

Before the Lotto Draw From 45 numbers you choose 6 numbers

Lotto Draw

Jackpot Match 6

Match 5 + Bonus

Match 5Trick

Match 4 + Bonus

Match 4

Match 3 + Bonus

Four distinguishable pairs of socks are in a drawer. If two socks are selected at random find the probability that a pair is chosen.And finally back to the Socks ProblemNo of favourable outcomes = (There are 4 pairs in the drawer)Total Number of Outcomes = (From 8 objects choose 2)

ARRANGMENTSSELECTIONSANDPROBABILITY

Cammie 2004NOTESCammie 2004NOTESCammie 2004NOTESCammie 2004NOTESCammie 2004NOTESCammie 2004NOTESCammie 2004NOTESCammie 2004NOTESCammie 2004NOTESCammie 2004NOTESCammie 2004NOTESCammie 2004NOTESCammie 2004NOTESCammie 2004NOTESCammie 2004NOTESCammie 2004NOTESCammie 2004NOTESCammie 2004NOTESCammie 2004Slide 25 keeping with the syllabus definition NOTESCammie 2004NOTESCammie 2004NOTESCammie 2004NOTESCammie 2003NOTESSlide 61trying to keep to a consistent font