Gill Sans Bold. General Mathematics Preliminary Course Stage 6. PB Probability. Value. Term number

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1 Gill Sans Bold General Mathematics Preliminary Course Stage 6 PB Probability II II ELIZABETH ELIZABETH AUSTRALIA AUSTRALIA Value Term number Revised 2003 MG.Prelim P

2 Acknowledgments This publication is copyright Learning Materials Production, Open Training and Education Network Distance Education, NSW Department of Education and Training, however it may contain material from other sources which is not owned by Learning Materials Production. Learning Materials Production would like to acknowledge the following people and organisations whose material has been used. Extract from General Mathematics Stage 6 Syllabus Board of Studies NSW, originally issued The most up-to-date version can be found on the Board's website at All reasonable efforts have been made to obtain copyright permissions. All claims will be settled in good faith. Writer: Editor: Revised: Illustrator: Cover illustration: Layout: Jim Stamell Teresa Ashton, Jan Harte Margaret Willard Jim Stamell, Thomas Brown, Barbara Gurney, Angela Truscott Thomas Brown Gayle Reddy Copyright in this material is reserved to the Crown in the right of the State of New South Wales. Reproduction or transmittal in whole, or in part, other than in accordance with provisions of the Copyright Act, is prohibited without the written authority of Learning Materials Production. Learning Materials Production, Open Training and Education Network Distance Education, NSW Department of Education and Training, Wentworth Rd. Strathfield NSW Revised August 2002

3 Contents Module overview... iii Outcomes... iv Indicative time... iv Course overview...v Icons... vi Formula sheet...vii viii PB1 The language of chance Part 1 The language of chance PB2 Relative frequency and probability Part 1 Relative frequency Part 2 Calculating probabilities Introduction i

4 ii PB Probability

5 Module overview The main focus of this unit is to provide the foundation in Probability required for this area of study and to use these skills to solve real and abstract problems. This module addresses the Probability area of study and has two units. PB1 The language of chance. In this unit, students learn to use the language of probability, count outcomes and describe the sample space of an event. PB2 Relative frequency and probability. The main focus of this unit is to compare the relative frequency of events and calculate probability. The relative importance of Probability in relation to the whole preliminary course is demonstrated in the Course Overview. Note that the order of listing is by topic, and is not necessarily a prescribed sequence in which students will study the whole course. Introduction iii

6 Outcomes Within the various parts of this module each of the following outcomes will be addressed to some extent. A student: P1 develops a positive attitude to mathematics and appreciates its capacity to provide enjoyment and recreation P2 applies mathematical knowledge and skills to solving problems within familiar contexts P3 develops rules to represent patterns arising from numerical and other sources P4 represents information in symbolic, graphical and tabular forms P10 performs simple calculations in relation to the likelihood of familiar events P11 justifies his/her response to a given problem using appropriate mathematical terminology Extract from General Mathematics Stage 6 Syllabus Board of Studies NSW, originally issued The most up to date version can be found on the Board's website at Indicative time Students should allocate up to 16 hours of the total 120 course hours to this area of study, allowing approximately up to 5 hours for the first part, and up to 10 hours for the second part, and leaving time for revision and set assignment tasks or examinations. iv PB Probability

7 Course overview Financial Mathematics FM1 Earning money FM2 FM3 Investing money Taxation Data Analysis DA1 Statistics and society DA2 DA3 DA4 Data collection and sampling Displaying single data sets Summary statistics Measurement M1 Units of measurement M2 M3 M4 Applications of area and volume Similarity of two dimensional figures Right angled triangles Probability PB1 The language of chance PB2 Relative frequency and probability Algebraic Modelling AM1 Basic algebraic skills AM2 Modelling linear relationships Introduction v

8 Icons The following icons are used within this module. The meaning of each icon is written beside it. There is an activity for you to do. You may need to collect data or make something. You need to use a computer for this activity. There are examples with solutions for you to work through. There is a review exercise for you to complete. For the latest version of the General Mathematics formula sheets go to _gen_02.pdf on the Board of Studies web site. vi PB Probability

9 2002 HIGHER SCHOOL CERTIFICATE EXAMINATION General Mathematics FORMULAE SHEET Area of an annulus A = p R -r R r Area of an ellipse A a b Area of a sector A q q p r 360 Arc length of a circle l q f d d m l = = = pab = = = = = = 2 2 ( ) q pr Simpson s rule for area approximation A ª h ( d f + 4 d m + d l ) 3 h = distance between successive measurements d = first measurement = = radius of outer circle radius of inner circle length of semi-major axis length of semi-minor axis 2 number of degrees in central angle number of degrees in central angle middle measurement last measurement Surface area Sphere Closed cylinder A = 2prh + 2pr r h Volume Cone Cylinder Pyramid Sphere r h A Sine rule radius perpendicular height 1 V = Ah V = pr 3 radius perpendicular height area of base Area of a triangle Cosine rule or = = = = = A 1 2 V = pr h 3 2 V = pr h a b c = = sin A sin B sinc 1 A = absinc c = a + b -2abcosC C a + cos = b - c 2ab 2 = 4pr 2

10 FORMULAE SHEET Simple interest I = Prn P r n Compound interest A = P 1 + r A P n r Future value ( A) of an annuity A M 1 + r 1 MÌ r Ó Present value ( N) of an annuity N or N = = = = = = = = = = = initial quantity percentage interest rate per period, expressed as a decimal number of periods ( ) n final balance initial quantity number of compounding periods percentage interest rate per compounding period, expressed as a decimal n Ï( ) - contribution per period, paid at the end of the period n ( ) - Ï 1 + r 1 MÌ n Ó r( 1 + r) A ( 1 + r) n Straight-line formula for depreciation S = V - Dn 0 S = salvage value of asset after n periods V0 = D = purchase price of the asset amount of depreciation apportioned per period n = number of periods Declining balance formula for depreciation S = V -r S r Mean of a sample x x x x n f x n fx f Formula for a z-score z s x - x s Gradient of a straight line m = = = = = = = = = = = ( ) Gradient intercept form of a straight line y = mx + b m = b = 0 1 Â Â Â gradient y-intercept Probability of an event n salvage value of asset after n periods percentage interest rate per period, expressed as a decimal individual score mean number of scores frequency standard deviation vertical change in position horizontal change in position The probability of an event where outcomes are equally likely is given by: number of favourable outcomes P(event) = total number of outcomes

11 Gill Sans Bold General Mathematics Preliminary Course Stage 6 PB1 The language of chance Part 1 The language of chance II II ELIZABETH ELIZABETH AUSTRALIA AUSTRALIA Value Term number Revised 2003

12 Gill Sans Bold Contents Introduction Talking chances More on language Sample space Number of outcomes for a multi-stage event More on combinations Terminology Exercise The language of chance Appendix Student evaluation Answers Part 1 The language of chance 1

13 2 PB1 The language of chance

14 Gill Sans Bold Introduction This part covers the basics of probability, an area of study of interest in our everyday lives. Specific content outcomes By the end of this Part, you will have been given opportunities to: order everyday events from the very unlikely to the almost certain use a list or table to identify the sample space determine whether or not the outcomes from an experiment are equally likely determine the outcomes for a multi-stage event use systematic lists to verify the total number of outcomes. Students of Distance Education Centres only: there is a page at the back for you to fill in when you have finished the work. Please say how easy/hard/interesting you find this work; ask relevant questions and return the comments with your work. Part 1 The language of chance 3

15 1.1 Talking chances Chance is a part of our everyday lives and probability is the study of chance events. Our language contains many words which talk about chance. There is a good chance the home team will win the game tomorrow. It is more than likely there will be a thunderstorm this afternoon. I think I have a chance of getting that job. Mary is hopeful she will go on an overseas holiday next year. I m almost certain Mrs Ashton will have a girl. Unless you study, the odds are against your passing the test. Sometimes we assign numbers to events according to how confident we are that they might happen. For example, we might say there is a 20% chance that the home team will win the game tomorrow. Be careful when numbers are used like this as they may not have any mathematical or statistical meaning! 4 PB1 The language of chance

16 Gill Sans Bold Our everyday language has many terms about the likelihood of an event occurring, for example: certain impossible odds against one in a million doubtful likely unexpected definite unlikely unsure sure perhaps possible odds on fluke improbable hopeful maybe even chance conceivable Buckley s chance lucky off-chance no way Exercise 1.1a 1 Choose two terms from the previous list which refer to: a an event which will happen b an event which will not happen c an event which may happen d an event just as likely to happen as not happen. Mathematically speaking An event which will never happen has a probability of zero. We say the event is impossible. An event which will definitely happen has a probability of one. We say the event is certain. Part 1 The language of chance 5

17 Here is a probability line from zero (impossible event) to one (certain event). As the probability of an event moves from zero towards one, the event becomes more likely. Impossible for the event to occur Example: the probability of a man becoming pregnant or 1 2 Event will definitely occur Example: the probability of finding sand on Bondi Beach. What do you think a probability of 0.5 or 1 2 means? Solution It means that an event with a probability of 0.5 is just as likely to happen as not to happen. 6 PB1 The language of chance

18 Gill Sans Bold Exercise 1.1b 1 Here are some probability statements used in everyday language: even chance odds against 5 50 surely impossible certain odds on doubtful off-chance one in a million On the diagram, write each of these terms in an oval, indicating with an arrow where the term might appear on the probability line. Even chance has been done for you. It's definitely a happening thing! Well... it could happen. even chance But then, it just as likely won't! It's just NOT going to happen! [There is another copy of this diagram in the Appendix.] 2 Imagine a drawer full of black socks. You enter the room in pitch darkness looking for socks. If you reach into the drawer and take out any sock, what is the probability that it will be: a a black sock? b a red sock? 3 a Now imagine that the drawer has an equal number of blue and black socks. If you reach into the drawer and take out a sock, what is the probability it will be: i a black sock? ii a blue sock? iii a red sock? iv a sock? Part 1 The language of chance 7

19 b What s the least number of socks you d need to take out of this drawer to make sure you have a matched pair? 4 Label the following events as possible, impossible, certain or uncertain. a Hope wins the lottery but has no lottery ticket. b Shirley wins the lottery after buying all the tickets. c Prudence wins the lottery after buying a ticket. d Burl wins the lottery after buying half the tickets. 5 Comment on the statement: Since it either rains or is fine, the probability of a fine day is What is meant by a one in 300 year flood? Explain. Investigation activity Meteorologists (weather forecasters) often make statements such as there is a 60% chance of rain tomorrow. Investigate how they arrive at such numbers. Internet address The Bureau of Meteorology has a website which has lots of information about the weather 8 PB1 The language of chance

20 Gill Sans Bold 1.2 More on language The Macquarie Dictionary defines chance as: the absence of any known reason why an event should turn out one way rather than another a possibility the probability of anything happening. Probability is defined as the likelihood or chance of something happening. Probability can be represented on a scale from zero (impossible) to one (certain) as you saw in the last section. The following diagrams may help you to answer questions in Exercise 1.2. The likelihood of an event becomes more certain and less unlikely from left to right. 0 impossible 1 2 The likelihood of an event becomes less certain and more unlikely from right to left. 1 certain fluke lucky hopeful off chance improbable unexpected conceivable unsure doubtful unlikely odds against perhaps possible one in a million incredible maybe likely credible almost certain 1 odds on expected sure thing safe bet good chance surely Part 1 The language of chance 9

21 a die The black suits: clubs and spades The red suits: hearts and diamonds a pair of dice Each suit has an: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen and King. The Ace can act as a one or the highest card. The Jack, Queen and King are called picture cards or royal cards. Exercise Use an everyday term to describe the possibility of the events listed below. a There will be rain somewhere in Australia this year. b All Tasmanians will move to the mainland. c No Australians will be smokers by the year d Tossing a six with a normal die. e Tossing a seven with a normal die. f Throwing an odd number with a normal die. g Throwing a total of seven with a normal pair of dice. h Choosing a black card from a full deck of cards. i Drawing a two from a stack of picture cards. j Picking a heart from a stack of black cards. 2 Explain the following expressions. a Once in a blue moon. b It was left to chance. c They have Buckley s chance of winning. d Hope against hope. e A snowflake s chance in hell. 10 PB1 The language of chance

22 Gill Sans Bold 3 A die is rolled. Arrange the following events from least likely to most likely. One event has been done for you. [A copy of this diagram is also in the Appendix. DEC students may need to return the diagram for marking.] 1 Event rolling an 8 less likely more likely Probability 1 2 rolling a number from 1 6 rolling an odd number rolling a 4 0 rolling an even number Part 1 The language of chance 11

23 1.3 Sample space An outcome is the result of an experiment or game. For example, getting a six is an outcome when a die is rolled. A sample space is the set of all possible outcomes. For example, when a die is rolled the sample space has six outcomes {1, 2, 3, 4, 5, 6}. When two unbiased coins are tossed simultaneously there are three possible results: two heads a head and a tail two tails. Notice that there are two ways of getting a head and a tail. This sample space contains four outcomes. Two outcomes (HT and TH) make up the event of getting one of each kind and the other two outcomes (HH and TT) make up the event getting two of a kind. We must be very careful in experiments like this to make sure we include all the possible outcomes in the sample space. In any given number of trials, the set of all possible outcomes is called the sample space. 12 PB1 The language of chance

24 Gill Sans Bold 1 List the sample space when a coin is tossed once. 2 List the sample space when two coins are tossed. 3 What is the sample space for choosing a vowel? Solutions 1 The sample space is S = {heads, tails} or S = {H, T} 2 The sample space is S = {HH, HT, TH, TT} 3 Sample space is S = {a, e, i, o, u} Exercise 1.3a 1 What is the sample space for each of these spinners? a b c 2 A bag contains six blue marbles and three white marbles. What is the sample space when a marble is selected? In a family there are two children. What is the sample space in terms of boys and girls? 4 In another family there are three children. What is the possible sample space now? 5 Three coins are tossed. List the sample space. 6 The first 12 letters of the alphabet are written on separate cards and put into a bag. One card is taken out. What is the sample space? Part 1 The language of chance 13

25 Large sample spaces For large sample spaces you use a list, table or tree diagram. 1 In a family there are four children. List the elements of the sample space using: a a list b a table c a tree diagram Solutions 1 There are 16 elements in the sample space. a Start the list with four boys and look at the pattern as you go down. BBBB BBBG BBGB BBGG BGBB BGBG BGGB BGGG GBBB GBBG GBGB GBGG GGBB GGBG GGGB GGGG b The table is drawn up by considering two arrangements of two children at a time, BB, BG, GB and GG. BB BG GB GG BB BBBB BBBG BBGB BBGG BG BGBB BGBG BGGB BGGG GB GBBB GBBG GBGB GBGG GG GGBB GGBG GGGB GGGG c 1st child 2nd child 3rd child 4th child sample space B G B G B G B G B G B G B G B G B G B G B G B G B G B G B G BBBB BBBG BBGB BBGG BGBB BGBG BGGB BGGG GBBB GBBG GBGB GBGG GGBB GGBG GGGB GGGG 14 PB1 The language of chance

26 Gill Sans Bold Exercise 1.3b 1 List the outcomes in the sample space when four coins are tossed? 2 The different colours of M&M s in a packet are: brown, green, orange, red, tan and yellow. Two M&M s are selected from a packet. What are the possible colour combinations? 3 Two dice are tossed. List the elements of the sample space. 4 The vowels [a, e, i, o, u] are each written on a separate card and put into a hat. Two cards are randomly drawn out. What is the sample space for this experiment if: a the first card is replaced before drawing out the second. b the first card is not replaced before drawing out the second. 5 Assume four people are asked to agree or disagree about some issue. What is the sample space if exactly two of the four people agree with the issue? 6 A set of five playing cards consists of the following: Jack of Hearts (a red card) Jack of Clubs (a black card) Eight of Hearts (a red card) Eight of Spades (a black card) Eight of Diamonds (a red card). Suppose one card was selected at random. a What colour is it most likely to be? b What suit (Hearts, Clubs, Spades, Diamonds) is it likely to be? c d True or false. An Eight has three times the chance of being chosen as a Jack. Two cards are drawn at random. How many elements would there be in the sample space? Part 1 The language of chance 15

27 1.4 Number of outcomes for a multi-stage event An older poker machine has a number of reels which can be made to spin by the use of a handle. Each reel has twenty symbols. The machine pays small prizes for certain frequent combinations and larger jackpot prizes for rare combinations. Note: in a club or hotel the types and numbers of symbols on each reel can be controlled and prizes calculated accordingly. The club or hotel will always win! Let us examine the possible outcomes for a poker machine with two reels of four symbols. First reel Second reel cactus cowboy skull cowboy horse guitar steer cowboy boots 1 How many possible arrangements are there? 2 How many ways can you get a horse/cowboy combination? 3 How many arrangements would be possible for a poker machine with: a 2 reels; 5 symbols on each b 2 reels; 9 symbols on each c 2 reels; 12 symbols on the first and 15 on the second 16 PB1 The language of chance

28 Gill Sans Bold d e f g 2 reels; 20 symbols on the first and 16 on the second 3 reels; 4 symbols on each 3 reels; 12, 15 and 18 symbols 3 reels; 10 symbols on the first and third and 15 on the second? Solutions 1 Draw a tree diagram to systematically list all possible combinations. Reel arrangements Display pairs There are 16 combinations. ÈNote: Í Î 16 = Two ways 2 reels means you multiply 2 numbers 4 symbols on each reel means you multiply 4 by 4 84 reels; 20 symbols on each Solution 3 a 5 5 = 25 b 9 9 = 81 Part 1 The language of chance 17

29 c = 180 d = e = 4 = 64 f = 3240 g = h = 20 = Note: a typical poker machine has 4 reels, each with 20 symbols. Since there are possible combinations, if you are trying to win a jackpot you probably have only a 1 in chance (or % chance) of getting it! Exercise 1.4 Calculate the number of possible combinations for the following number of reels and symbols: 1 4 reels, 5 symbols on each 2 4 reels, 9 symbols on each 3 4 reels, 20 symbols on each 4 5 reels, 12 symbols on each 5 5 reels, 15 symbols on each 6 5 reels, 20 symbols on each 18 PB1 The language of chance

30 Gill Sans Bold 1.5 More on combinations In the previous section you found the number of outcomes for a multi-stage event by multiplying the number of choices at each stage. For example: a poker machine with three reels of 12, 15 and 20 symbols has = 3600 possible combinations. choosing 2 different vowels from the set {a, e, i, o, u} has 5 4 = 20 possible combinations. Note: for the poker machine, choosing a symbol on one reel does not affect the choice for the other reels. However, for the vowels, the first choice impacts on the second choice. There is one less choice for the second vowel. In how many ways can three letters be placed in three envelopes? Solution Method 1 Use a table First envelope Letter 1 Second envelope Letter 2 Letter 3 Third envelope Letter 3 Letter 2 Letter 2 Letter 1 Letter 3 Letter 3 Letter 1 Letter 3 Letter 1 Letter 2 Letter 2 Letter 1 Part 1 The language of chance 19

31 Method 2 Use a tree diagram E1 E2 E3 L1 L2 L3 L2 L3 L1 L3 L1 L2 L3 L2 L3 L1 L2 L1 Note: since you do not actually have to list all outcomes in the sample space, there is a simpler way of finding the number of outcomes. In this example, once a letter is placed into an envelope it cannot be used again. So, there are three choices for the first envelope, two choices for the second envelope and only one for the third envelope. No. of ways = = 6. In the following exercises you will need to decide whether choices are affected or not. Exercise In how many ways can you place four letters in four separate envelopes? 2 At a school dance there are six couples. How many different male-female pairs are possible for a particular dance? 3 At a restaurant the menu has: four entrées ten main courses and five desserts. How many different three-course meals are possible? 20 PB1 The language of chance

32 Gill Sans Bold 4 Braille is a system of touch reading and writing for blind persons in which raised dots represent the letters of the alphabet. Each Braille cell has six dots which can be raised or not raised. Two possible combinations are shown. How many possible combinations are there? 5 The original Morse code used short electrical signals (dots, ) and long electrical signals (dashes, ). For example, the letter E consists of one dot ( ) but the letter P consists of two dots and two dashes ( ). Each letter has up to four dots or dashes. a How many possibilities are there if: i one symbol (dot or dash) is allowed ii two symbols are allowed iii three symbols are allowed iv four symbols are allowed. b How many dot/dash combinations are possible? c d Why is a maximum of four dot/dash combinations sufficient to represent the alphabet? The American Morse code uses up to five dot/dash combinations to represent the letters of the alphabet. What is the total number of combinations possible? 6 Computers use binary digits (or bits) instead of decimal digits. Bits have two possible values: 0 or 1. If there are eight bits in a byte, how many different bytes are possible? 1 byte = 8 bits 7 In Australia, postcodes consist of four digits, with postcodes in NSW beginning with two. a How many possible NSW postcodes are there? b Australian postcodes begin with 0,2,3,4,5,6 or 7. How many postcodes are possible in Australia? c In Tasmania postcodes run from 70 to 74. How many postcodes does Tasmania have? 2224 Part 1 The language of chance 21

33 8 Most NSW car number plates consist of three letters and three numbers a How many number plates are possible? b What do you think could be done if NSW needs more number plates? DTI About 40 years ago telephone numbers in Sydney consisted of two letters followed by four numbers. The letters used were: A B F J L M U W X Y a How many combinations were possible? b Today most telephone numbers consist of up to eight digits. Why did the system change? Investigation activity 1 Investigate whether all possibilities of raised dots are used in Braille. 2 Investigate how numbers and punctuation marks are represented in Morse code. 3 Collect a number of statements from various media involving the language of probability. Good sources are newspapers, magazines, radio, television or the Internet. Write a comment for each statement. 22 PB1 The language of chance

34 Gill Sans Bold Terminology Do you understand the following words or expressions? Look back in your notes, if necessary, and give your explanations here. chance probability sample space outcome multi-stage event Part 1 The language of chance 23

35 24 PB1 The language of chance

36 Gill Sans Bold Exercise -The language of chance 1 Name a game which relies totally on chance. Explain why skill would have no influence on the outcome of such a game. 2 Write down two sentences of your own which show that chance and possibilities are used in everyday language. 3 What is meant by the following terms: a b one in a million c odds against. 4 Comment on the use of numbers in this statement: "I think St. George has a 90% chance of winning the premiership this year". 5 Three coins are tossed simultaneously. In how many ways can you get: a two tails and one head? b two heads or three heads? 6 In a bag are four coloured marbles: red, yellow, blue, green. Two marbles are drawn at random. a How many elements in this sample space? b List the elements in the sample space. 7 What is the total number of ways you can place 5 letters in five different envelopes? Part 1 The language of chance 25

37 8 In one bag there are 26 cards printed with a letter of the alphabet. Another bag has cards with each of the letters of the word PROBABILITY. A student randomly selected one card from each bag and found that the letter B was on each card. He commented that the chance of selecting this letter must have been the same. Do you agree? Explain. 9 A poker machine has three reels with the following numbers of symbols. Symbol First reel Second reel Third reel a b How many symbols altogether are on each reel? How many possible arrangements are there? c In how many ways can three be spun up? d Which is more likely to occur: or? e How many ways can you get in this order? 10 Semaphore flag signalling is a system based on the waving of a pair of hand-held flags in a particular pattern. There are eight possible positions to hold each flag. How many signals are possible? 11 A, B, and C are towns. There are three ways to get from A to B and four ways to get from B to C. How many possible ways are there to get from A to C? 26 PB1 The language of chance

38 Gill Sans Bold Appendix Copy of diagram for question 1 in Exercise 1.1b It's definitely a happening thing! Well... it could happen. even chance But then, it just as likely won't! It's just NOT going to happen! Part 1 The language of chance 27

39 Copy of diagram question 3 Exercise Event rolling an 8 less likely more likely Probability 1 2 rolling a number from 1 6 rolling an odd number rolling a 4 0 rolling an even number 28 PB1 The language of chance

40 Gill Sans Bold Student evaluation When you have finished this unit of work see if you can do these things. Tick if you can do them with confidence: Order everyday events from the very unlikely to the almost certain Use a list or table to identify the sample space can do with confidence need more help Determine whether or not the outcomes from an experiment are equally likely Determine the outcomes for a multi-stage event Use systematic lists to verify the total number of outcomes. Ask for further help with any you feel unsure about. Please write your questions and any other comments here. Part 1 The language of chance 29

41 30 PB1 The language of chance

42 Gill Sans Bold Answers Exercise 1.1a 1 a certain, definite b impossible; Buckley s chance; no way c Many of the words in the list mean that something may happen. d even chance; Exercise 1.1b 1 Note: except for the terms which relate to the probabilities of 0, 1 and, 1 2, where you place a term along the line is personal. The diagram below shows how one person feels about the terms. It's definitely a happening thing! odds on off chance Well... it could happen. certain one in a million even chance But then, it just as likely won't! surely It's just NOT going to happen! impossible doubtful odds against 2 a 1 b 0 3 a i 1 2 ii 1 2 iii 0 iv 1 b Three since the first two you take out may or may not be a pair but the third one has to match at least one of them 4 a impossible b certain c possible d possible Part 1 The language of chance 31

43 5 While there are two outcomes here (ignoring for the moment other possibilities) each outcome is not equally likely. For NSW at least, it is more likely not to rain than to rain so probability is definitely not The chance of the flood is very unlikely. If there is any mathematical basis to the number 300 years, it might be that over a very long period of time the flood would occur on average this often. This by no means indicated it floods once every 300 years. Exercise Note: except for the terms which relate to the probabilities of 0, 1 and 1 2, the terms you choose are open to personal interpretation. a certain b unlikely c doubtful d possible e impossible f even chance g possible h even chance i impossible j impossible 2 (Use a dictionary, or other source, to find the answers.) 3 1 Event rolling an 8 less likely more likely Probability 1 2 rolling a number from 1 6 rolling an odd number rolling a 4 0 rolling an even number 32 PB1 The language of chance

44 Gill Sans Bold Exercise 1.3a 1 a {1, 2, 3, 4} b {1, 2, 3, 4, 5, 6} c {1, 2, 3} 2 {B, B, B, B, B, B, W, W, W} 3 {BB, BG, GB, GG} Note: order is important here. For example, BG implies a younger boy whereas GB implies an older boy. 4 {BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG} 5 {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} 6 {A, B, C, D, E, F, G, H, I, J, K, L} Exercise 1.3b 1 {HHHH, HTHH, THHH, TTHH, HHHT, HTHT, THHT, TTHT, HHTH, HTTH, THTH, TTTH, HHTT, HTTT, THTT, TTTT} 2 {BB, BG, BO, BR, BT, BY, GB, GG, GO, GR, GT, GY, OB, OG, OO, OR, OT, OY, RB, RG, RO, RR, RT, RY, TB, TG, TO, TR, TT, TY, YB, YG, YO, YR, YT, YY} 3 {1 1, 1 2, 1 3, 1 4, 1 5, 1 6, 2 1, 2 2, 2 3, 2 4, 2 5, 2 6, 3 1, 3 2, 3 3, 3 4, 3 5, 3 6, 4 1, 4 2, 4 3, 4 4, 4 5, 4 6, 5 1, 5 2, 5 3, 5 4, 5 5, 5 6, 6 1, 6 2, 6 3, 6 4, 6 5, 6 6} 4 a {aa, ae, ai, ao, au, ea, ee, ei, eo, eu, ia, ie, ii, io, iu, oa, oe, oi, oo, ou, ua, ue, ui, uo, uu} b {ae, ai, ao, au, ea, ei, eo, eu, ia, ie, io, iu, oa, oe, oi, ou, ua, ue, ui, uo} (did you realise the sample space was the same as before except you can t have aa, ee, ii, oo, or uu?) 5 {YYNN, NNYY, YNYN, NYNY, YNNY, NYYN} (Y means agree; N means disagree) 6 a red b hearts c False. Eight has a 3 5 chance and Jack has a 2 5 chance. d 5 4 = 20, that is, five choices for first leaving four choices for second card. Exercise a 5 = 625 b 9 = 6561 c 20 4 = d 12 5 = e f Part 1 The language of chance 33

45 Exercise = = = =64 5 a i 2 ii 2 2 = 4 iii 2 3 = 8 iv 2 4 = 16 b = 30 c 26 letters in the alphabet and 30 possibilities d = = a = Note: 10 (0 to 9) ways to choose each digit b 7000 c a = b There are lots of possibilities: start again using different coloured plates; change the order of letter/number combinations; add more letters or numerals. ( ) ( ) = 9 a b Increasing population makes demands for more phones. More digits allow for more combinations. 34 PB1 The language of chance

46 Gill Sans Bold General Mathematics Preliminary Course Stage 6 PB2 Relative frequency and probability Part 1 Relative frequency II II ELIZABETH ELIZABETH AUSTRALIA AUSTRALIA Value Term number Revised 2003

47 Contents Introduction Experiment flipping coins Experiment dice and cards Relative frequency Experiments with relative frequency Talking it through...17 Terminology...25 Exercise Relative frequency...27 Student evaluation...29 Appendix...31 Answers...33 Part 1 Relative frequency 1

48 2 PB2 Relative frequency and probability

49 Gill Sans Bold Introduction This Part covers relative frequency of events and calculate probability. Specific content outcomes By the end of Part 1, you will have been given opportuniites to: perform simple experiments to obtain relative frequencies from recorded data estimate the relative frequencies of events from recorded data use relative frequencies to obtain approximate probabilities illustrate the results of experiments through statistical graphs and displays. Students of Distance Education Centres only: send in pages apply to students studying this course through Distance Education. Other students need not return these pages. There is a page at the back for you to fill in when you have finished the work. Please say how easy/hard/interesting you find this work; ask relevant questions and return the comments with your work. Part 1 Relative frequency 3

50 1.1 Experiment flipping coins A probability experiment usually involves repetition of a single planned action, such as flipping coins. The coins on the right all show heads. Those on the left show tails. Doug thinks that the chance of flipping a head or a tail should be the same. He flips the coin ten times and gets the following: He records his results on a tally sheet. Heads Tails 4 PB2 Relative frequency and probability

51 Gill Sans Bold Exercise a How many heads did Doug toss? b How many tails were flipped? c Does this support what Doug thinks the results might be? Why? 2 Doug tosses the coin another ten times and gets five heads and five tails. a b Does this support what Doug thinks the results might be? Give a reason for your answer. Add the score for the second round of ten tosses to the tally sheet on the previous page. 3 Doug tosses the coin: ten more times and gets seven heads and three tails another ten times and gets three heads and seven tails another ten times and gets six heads and four tails a Add these tosses to the tally sheet. b How many heads (H) and tails (T) did he get in the first 50 tosses? 4 Here are the next 50 tosses: a Record the results on the tally sheet. b Give the results of coin tosses c What are the cumulative results of the 100 coin tosses? d How do these results fit with Doug s expectation? Part 1 Relative frequency 5

52 Investigation activity You will need a 20c coin for this exercise. 1 a Flip your coin ten times and record the number of heads and tails on the tally sheet. b Repeat for tosses c Continue in sets of ten until you reach 100 tosses. Tally Cumulative scores Heads Tails Tosses Heads Tails Use the cumulative scores to draw a column graph Consider the number of heads and tails for the first 20 tosses, 50 tosses, 80 tosses, 100 tosses. Comment on your results as the sample size increases. 6 PB2 Relative frequency and probability

53 Gill Sans Bold 1.2 Experiment dice and coins The first part of this section looks at dice and the second part looks at cards. Dice You will need a die and a cup to shake it for the first exercise. Exercise 1.2a 1 Examine your die to answer the following. a What number is on the face opposite the six? b What number is on the face opposite the two? c What number is on the face opposite the four? d What do opposite faces on a die add up to? 2 a Answer true or false to the following statements. b i ii iii The chance of rolling a six is the same as rolling any other number. The chance of rolling an even number is greater than the chance of rolling an odd number. There are six possible results when a die is rolled. What do you think will happen if a die is biased, that is, not evenly weighted? 3 a If you rolled a fair die 60 times, how many times would you expect to roll a three? Give a reason for your answer. Part 1 Relative frequency 7

54 b Would this definitely happen if you did roll a die 60 times? Why? 1 Roll a die and record the result on the tally sheet. For example, if you roll a four, put a stroke in the four column. 2 a Repeat another 59 times, marking the tally sheet each roll. b Total the results for the 60 rolls and put in the Score 1 60 rolls row. Uppermost face Tally Score 1-60 rolls Score rolls 3 a Roll the die another 60 times, marking the tally sheet after each roll. [Use a different coloured pen for rolls ] b Total all scores and write the results in the Score rolls row. 4 Represent your results in a column graph. 5 What did you expect your graph to look like? Why? 6 If you combined your 120 rolls with those of five other students, what might you expect the results of 600 rolls to be? 8 PB2 Relative frequency and probability

55 Gill Sans Bold Solutions 4 Here is a possible column graph. Each student s graph will vary depending on the results they get. This part of the graph shows that a four was rolled 12 times in the first 60 rolls 25 times in the120 rolls 30 cumulative score rolls 120 rolls 60 rolls 120 rolls 60 rolls 120 rolls 60 rolls 120 rolls 60 rolls 120 rolls 60 rolls 120 rolls uppermost face 5 You would expect the graph to have six columns of equal height since each number on the die is expected to come up the same number of times. 6 You would expect to get 100 rolls of each number. Cards In a normal deck (or pack) of cards there are 26 red cards and 26 black cards. There are four suits, each containing 13 cards. heart diamond spade club red cards black cards Note: you need to know the makeup of a deck of cards. Part 1 Relative frequency 9

56 Exercise 1.2b You will need a normal deck of 52 cards for this exercise. [no jokers] 1 a Sort the cards into red and black. i How many black cards are there? ii How many red cards are there? b Answer true or false to the following statements. i There s a chance of drawing a red card from a full deck. ii There s an even chance of drawing a black card from a full deck. 2 a Sort the cards into the 4 suits. i How many hearts are there? ii How many diamonds are there? iii How many clubs are there? iv How many spades are there? b Answer true or false to the following statements. i There s a chance of drawing a diamond from a full deck. ii There s an even chance of drawing a club from all the black cards. 3 Collect all the cards which make up any one of the four suits. a b c d How many cards make up the suit? If you count the Ace as one, how many numbered cards make up the suit? How many other cards are in the suit? Answer true or false to the following statements. i ii There s a better than even chance of drawing a numbered card from a suit than a picture card. You re more likely to draw a numbered card than a picture card from a full deck. 4 In some card games, an Ace is low and has a value of one. In other games an Ace is high. What do you think Ace high means? 10 PB2 Relative frequency and probability

57 Gill Sans Bold 1.3 Relative frequency You expect to get 50 heads and 50 tails if a coin is tossed 100 times. You say the two outcomes are equally likely. If you actually toss a coin 100 times you might get 47 heads and 53 tails or perhaps 55 heads and 45 tails. Each time you do the experiment you might probably get slightly different results. The experimental result of an outcome is called the relative frequency. relativefrequency = number of times event occurs total number of trials The relative frequency of outcome 47 heads out of 100 tosses is or 47%. The first 100 Smiths from a local telephone directory are in the Appendix. 1 How many of these 100 telephone numbers end in a 6? 2 What is the relative frequency of the last digit being a six? 3 How many telephone numbers ending in a 6 do you expect in 5000 listings? Solutions 1 Ten numbers out of the 100 in this sample end in a six. [Check by counting] 2 Relative frequency of a 6 = = 1 10 or 10% 3 We can use our sample result to get a reasonable estimate. 10% of 5000 = = 500 We expect 500 numbers to end in a 6. Note: this is only an estimate. We don t know if there will be exactly 500 numbers ending in a 6 but we expect close to this number. Part 1 Relative frequency 11

58 Exercise a How many possibilities are there for the last digit of any telephone number? List them. b c d How many telephone numbers in the Smith sample of telephone numbers end with the digit nine? What percentage of the Smith telephone numbers end with the digit nine? Use this information from the sample to estimate how many telephone numbers end with the digit 9 in a listing of: i 5000 ii iii iv a million. Investigation activity You will need a copy of the white pages telephone directory for this exercise. 1 Open to any page and choose one listing. This is your first listing. Now count off, say,100 listings and mark the end one. It s okay if you go over to the next page. 2 a How many listing in your random sample? b c d How many end in nine? What is the relative frequency of the outcome a number ending in nine? Use the information from the sample to estimate how many telephone numbers end with a nine in a listing of: i 100 ii iii iv a half million 12 PB2 Relative frequency and probability

59 Gill Sans Bold 3 a Select another digit from 0 to 8. b c d How many of your sample numbers end in this digit? What is the relative frequency of this digit in your sample? Use the information to estimate how many telephone numbers end with this digit in a listing of: i 100 ii ii Part 1 Relative frequency 13

60 1.4 Experiments with relative frequency Relative frequency is the experimental value and probability is the theoretical or expected value where: relativefrequency = number of times event occurs total number of trials In an experiment where 100 coins are tossed, the relative frequency of tossing a head might be whereas in another it might be The average of all these relative frequencies would equal the predicted probability of 50 over a large number of trials. 100 Investigation activity For this activity you will need a matchbox (or similar). When a matchbox is tossed there are three ways it can land: on a base on a side on a end. Base side End 14 PB2 Relative frequency and probability

61 Gill Sans Bold 1 How many of each outcome would you expect in 50 tosses? 2 Toss your matchbox 50 times and record your results. Outcome Tally Frequency Relative Frequency Total trials = 50 3 Calculate the relative frequency of each outcome. 4 How do these relative frequencies compare with your guesses? Note: Conducting an experiment is about guessing the outcome and then deciding whether your guess is correct or not. Part 1 Relative frequency 15

62 Investigation activity For this activity you will need drawing pins (or similar objects). A drawing pin can land in one of two ways: Head Side Write down what you think the relative frequency for each of these outcomes might be. Toss one drawing pin 100 times. [To save time you could toss 10 pins ten times or 20 pins five time]. Record your results for 100 tosses of the drawing pin. Outcome Tally Frequency Relative Frequency Total trials = 100 Calculate the relative frequency of each outcome. How do these relative frequencies compare with your guesses? 16 PB2 Relative frequency and probability

63 Gill Sans Bold 1.5 Talking it through An important part of this course is learning to apply mathematical knowledge about probability to everyday statements and events. Suppose the letters in the word M A T H E M A T I C S were written on tiles, placed in a bag and one tile was drawn out. A I C M T H T E S A M M A T There are 11 tiles to choose from but the letters the chance of choosing one of these letters is twice as likely. appear twice so For example, the probability of choosing the letter M is 2 while the 11 probability of choosingthe letter C is Another important aspect of probability is randomness. If you reached inside the bag and chose a tile, you wouldn t know which letter it was until you looked at it if each tile was identical in size and feel. If however, you knew that the tile with the letter H on it was chipped you could pick it every time and choosing a letter would no longer be random. H????? Here is a question to ponder????? Why do you think many television and radio competitions ask you to send in your name and address written on a standard sized envelope? Why not any sized envelope? Part 1 Relative frequency 17

64 Exercise 1.5a 1 a Think about each of these probability statements then discuss them with someone. b Write your response to each statement.. Kwan s statement As there are twenty six letters in the English alphabet, the probability that a person s name starts with X is one in twenty six. Kiri s statement If I toss a fair coin six times and get six tails, the probability of getting a head for the next toss is greater than an even chance. Kurt s statement Since traffic lights can be red, amber or green, the probability that a light is red at any instant is one in three. $ Ken s statement There are tickets sold for each $2 jackpot lottery. My chances of winning a prize will increase if I buy a ticket in every lottery. 18 PB2 Relative frequency and probability

65 Gill Sans Bold 2 There is something wrong with the reasoning in each of the following statements. Identify it and comment critically on each statement. a b Mrs Jones has three children; all boys. She is now pregnant and is expecting another baby. This one is sure to be a girl. Two teams, Alpha and Omega, are playing a game of soccer. Since the results can be win, lose or draw the chance Omega wins is 1 3. c d e f g h i When two coins are tossed, the result can be: two heads two tails one head and one tail. Alexis concludes that the chance of each outcome is the same. At a swimming carnival there are eight lanes and Myra is in one of the lanes. Her chance of winning a race is 1 8. Barry is also swimming at the carnival. There are prizes for first, second and third place. Since there are four outcomes (first, second, third, no prize), his chance of winning a prize is 3 4. When I apply for a job I could either get it or not get it. My chances of getting the job are I put my money on Knackery Lad in the fifth race at the local racecourse. There are seven other horses running. My chance of getting a place (first, second or third) is 3 8. When you throw two dice you have the same chance of getting a seven as you have of getting a 12. In my next Chemistry test there are two outcomes: pass or fail. My chance of passing is only 50%. Note: if you have difficulty with any of these, ask yourself these questions. What assumptions am I making? Are the events random? Does each event have the same likelihood of happening? Is something other than chance influencing the outcome? Part 1 Relative frequency 19

66 1 The table shows the frequency per 1000 letters of each letter of the alphabet in normal English text. E 131 D 38 W 15 T 105 L 34 B 14 A 82 F 29 V 9 O 80 C 28 K 4 N 71 M 25 X < 2 R 68 U 25 J > 1 I 63 Y 20 Q > 1 S 61 G 20 Z < 1 H 53 P 20 a In an essay of letters, how many N s do you expect? b A letter was chosen at random from a normal page of writing. What is the chance it is the letter Y? Give your answer as: i ii iii a fraction a decimal a percentage. c In a short story of letters, how many A s should you find? d If S occurred 92 times in 1480 letters, what is its relative frequency? e What percentage of E s would you expect in an essay of 5600 words? Solution 1 a Number of N's = b i 20 1 = = ii 0.02 iii 2% 20 PB2 Relative frequency and probability

67 Gill Sans Bold 82 c Number of A's = = d Relative frequency of S = 92 = ª 6. 2% 1480 e Percentge of E's = % =. % [Note: the percentage of E s is the same for any number of letters] Note: over a very large number of trials the relative frequency of an outcome is equal to the probability (expected value) of that outcome. From the given table, the probability that a letter picked at random from normal English prose is 61 the letter S is % = 6. 1 %. In part d of the example above, the relative frequency of S is 6.2%. This is slightly above the expected value. Exercise 1.5b 1 Using the letter frequency table on the previous page: a b c d e how many times would you expect to see the letter R in every 1000 letters? if a letter is chosen at random from a page in a novel, what would be the probability (as a percentage) of it being the letter: i T? ii G? iii K? How many letters have less than a 0.5% chance of being chosen at random from a page in a novel? Which letter has a 3.8% chance of being chosen at random from a page in a novel? In an essay of words, the average word length is five letters. i How many letters in the essay? ii How many times would you expect to see the letter M? Part 1 Relative frequency 21