Probability PDF

10 Probability LEARNING SEQUENCE 10.1 Overview...............................................................................................................................................................538 10.2 Theoretical probability..............................................................

10 Probability LEARNING SEQUENCE 10.1 Overview...............................................................................................................................................................538 10.2 Theoretical probability.....................................................................................................................................544 10.3 Experimental probability.................................................................................................................................551 10.4 Venn diagrams and two-way tables...........................................................................................................559 10.5 Two-step experiments.....................................................................................................................................576 10.6 Review................................................................................................................................................................... 588 10.1 Overview Why learn this? New technologies which predict human behaviour are all based on probability. On 11 May 1997, IBM’s supercomputer Deep Blue made history by defeating chess grandmaster Gary Kasparov in a six-game match under standard time controls. This was the first time that a computer had defeated the highest ranked chess player in the world. Deep Blue won by evaluating millions of possible positions each second and determining the probability of victory from each possible choice. This application of probability allowed artificial intelligence to defeat a human who had spent his life mastering the game. Combining advanced computing with probability is no longer used only to play games. From the daily convenience of predictive text, through to assisting doctors with cancer diagnosis, companies spend vast amounts of money on developing probability-based software. Self-driving cars use probability functions to predict the behaviour of both other cars on the road and pedestrians. The more accurate the predictive functions, the safer self-driving cars can become. Even politicians use campaign data analysis to develop models that produce predictions about individual citizens’ likelihood of supporting specific candidates and issues, and the likelihood of these citizens changing their support if they’re targeted with various campaign interventions. As technology improves, so will its predictive power in determining the likelihood of certain outcomes occurring. It is important that we study and understand probability so we know how technology is being used and the impact it will have on our day-to-day life. Where to get help Go to your learnON title at www.jacplus.com.au to access the following digital resources. The Online Resources Summary at the end of this topic provides a full list of what’s available to help you learn the concepts covered in this topic. Fully worked Video Interactivities solutions eLessons to every question Digital eWorkbook documents 538 Jacaranda Maths Quest 9 Exercise 10.1 Pre-test Complete this pre-test in your learnON title at www.jacplus.com.au and receive automatic marks, immediate corrective feedback and fully worked solutions. 1. MC A six-sided die is rolled, and the number uppermost is noted. The event of rolling an even number is: A. {1, 2, 3, 4, 5, 6} B. {0, 1, 2, 3, 4, 5, 6} C. {2, 4, 6} D. {1, 3, 5} E. {2, 4, 5, 6} 2. The coloured spinner shown is spun once and the colour noted. Written in its simplest form, state the probability of spinning: a. an orange and a blue b. an orange or a pink. 3. A coin is tossed in an experiment and the outcomes recorded. Outcome Heads Tails Frequency 72 28 a. Identify how many trials there were. b. Calculate the experimental probability for tossing a Tail, giving your answer in simplest form. c. Calculate the theoretical probability for tossing a Tail with a fair coin, giving your answer in simplest form. d. In simplest form, calculate the difference in this experiment between the theoretical and experimental probabilities for tossing a Tail. 4. A random number is selected from: 𝜉 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} Calculate the exact probability of selecting a number that is a multiple of 3, giving your answer in simplest form. 5. Identify whether the following statement is True or False. If a large number of trials is conducted in an experiment, the relative (experimental) frequency of each outcome will be very close to its theoretical probability. 6. MC Consider the Venn diagram shown. A B ξ If 𝜉 = {numbers between 1 and 20 inclusive}, identify which of the following set B is equal to. 5, 19, A. B = {multiples of 3 numbers between 1 and 20} 2 7, 13, 1 B. B = {prime number 1} 11, 17 C. B = {even numbers between 1 and 20} 3 D. B = {odd numbers between 1 and 20} 9, 15 E. B = {prime numbers between 1 and 20} 6, 12, 18 C TOPIC 10 Probability 539  7. MC The information presented in the Venn diagram can be shown on a two-way frequency table as: A B ξ 6 3 5 A. B. A A′ Total A A′ Total B 5 3 8 B 3 6 9 B′ 0 6 6 B′ 5 0 5 Total 5 9 14 Total 8 6 14 C. D. A A′ Total A A′ Total B 3 5 8 B 3 2 5 B′ 6 0 6 B′ 3 3 6 Total 9 5 14 Total 6 5 11 E. A A′ Total B 3 3 6 B′ 2 3 5 Total 8 6 11 8. MC If 𝜉={numbers between 1 and 20 inclusive}, identify the complement of the event A={multiples of 5 and prime numbers}. A. {1, 4, 6, 8, 9, 12, 14, 16, 18} B. {5, 10, 15, 20} C. {2, 3, 5, 7, 11, 13, 17, 19} D. {2, 3, 5, 7, 10, 11, 13, 15, 17, 19, 20} E. {4, 6, 8, 9, 12, 14, 16, 18} 9. The jacks and aces from a deck of cards are shuffled, then two cards are drawn. Calculate the exact probability, in simplest form, that two aces are chosen: a. if the first card is replaced b. if the first card is not replaced. 10. The Venn diagram shows the results of a survey where students were asked to indicate whether they prefer drama (D) or comedy C D ξ (C) movies. In simplest form, determine the probability that a student selected at random prefers drama movies but does not like comedy. 22 6 20 2 540 Jacaranda Maths Quest 9 11. From a bag of mixed lollies, students can choose from three different types of lollies: red frogs, milk bottles or jelly babies. The bag contains five of each type of lolly. If Mahsa chooses two lollies without looking into the bag, calculate the probability that she will choose two different types of lollies. 12. The students in a class were asked about their sport preferences — whether they played basketball or tennis or neither. The information was recorded in a two-way frequency table. Basketball (B) No basketball (B′ ) Tennis (T) 25 20 No tennis (T′ ) 10 5 a. MC The student sport preferences recorded in the two-way frequency table is represented on a Venn diagram as: A. B. B T ξ B T ξ 10 25 20 10 25 20 5 C. D. B T ξ B T ξ 20 25 10 20 5 10 5 5 E. B T ξ 10 5 10 25 b. If one student is selected at random, calculate the probability that the student plays tennis only, correct to 2 decimal places. TOPIC 10 Probability 541  13. Match the following Venn diagrams to the correct set notation for the shaded regions. Venn diagram Set notation a. A. A ∩ B A B b. B. A′ A B ξ c. C. A ∩ B′ A B ξ d. D. A ∪ B A B ξ 14. A survey of Year 8, 9 and 10 students asked the students to choose dinner options for their school camp. 542 Jacaranda Maths Quest 9  Year level Lasagna Stir-fry 8 82 75 9 67 90 10 89 45 Total 240 210 Calculate the following probabilities. a. The probability that a randomly selected student chose lasagne b. The probability that a randomly selected student was in Year 10 c. The probability that a randomly selected student was in Year 8 and chose stir-fry d. The probability that a randomly selected student who chose lasagna was in Year 9 15. MC A bag contains three blue balls and two red balls. A ball is taken at random from the bag and its colour noted. Then a second ball is drawn, without replacing the first one. Identify the tree diagram that best represents this sample space. A. 3 B. 2 – B – B 5 5 3 B 2 B – – 5 5 2 3 – R – R 5 5 3 B 2 B – – 5 5 2 3 – R – R 5 5 2 3 – R – R 5 5 C. 1 D. 2 – B – B 3 4 1 B 3 B – – 3 5 1 2 – R – R 2 4 1 B 3 B – – 3 4 1 2 – R – R 2 5 1 1 – R – R 2 4 E. 1 – B 3 1 B B – 1 2 – 1 3 – B 3 1 B – 3 1 – R B 2 1 – 1 3 – B 3 TOPIC 10 Probability 543  10.2 Theoretical probability LEARNING INTENTION At the end of this subtopic you should be able to: determine whether events are impossible, unlikely, likely or certain use key probability terminologies such as trials, outcomes, sample space and events calculate the theoretical probability of an event. 10.2.1 The language of probability eles-4877 The probability of an event is a measure of the likelihood that the event will take place. If an event is certain to occur, then it has a probability of 1. If an event is impossible, then it has a probability of 0. The probability of any other event taking place is given by a number between 0 and 1. An event is likely to occur if it has a probability between 0.5 and 1. An event is unlikely to occur if it has a probability between 0 and 0.5. Unlikely Even Likely chance Impossible Certain 0 0.25 0.5 0.75 1 0% 50% 100% WORKED EXAMPLE 1 Placing events on a probability scale On the probability scale given, insert each of the following events at appropriate points. 0 0.5 1 a. You will sleep tonight. b. You will come to school the next Monday during a school term. c. It will snow in Victoria this year. THINK WRITE/DRAW a. 1. Carefully read the given statement and label a. a its position on the probability scale. 0 0.5 1 2. Write the answer and provide reasoning. Under normal circumstances, I will certainly sleep tonight. b. 1. Carefully read the given statement and label b. b its position on the probability scale. 0 0.5 1 544 Jacaranda Maths Quest 9  2. Write the answer and provide reasoning. It is very likely but not certain that I will come to school on a Monday during term. Circumstances such as illness or public holidays may prevent me from coming to school on a specific Monday during a school term. c. 1. Carefully read the given statement and label c. c (Summer) c (Winter) its position on the probability scale. 0 0.5 1 2. Write the answer and provide reasoning. It is highly likely but not certain that it will snow in Victoria during winter. The chance of snow falling in Victoria in summer is highly unlikely but not impossible. 10.2.2 Key terms of probability eles-4878 The study of probability uses many special terms that must be clearly understood. Chance experiment: A chance experiment is a process, such as rolling a die, that can be repeated many times. Trial: A trial is one performance of an experiment to get a result. For example, each roll of the die is called a trial. Outcome: The outcome is the result obtained when the experiment is conducted. For example, when a normal six-sided die is rolled the outcome can be 1, 2, 3, 4, 5 or 6. Sample space: The set of all possible outcomes is called the sample space and is given the symbol 𝜉. For the example of rolling a die, 𝜉 = {1, 2, 3, 4, 5, 6}. Event: An event is the favourable outcome of a trial and is often represented by a capital letter. For example, when a die is rolled, A could be the event of getting an even number; A = {2, 4, 6}. Favourable outcome: A favourable outcome for an event is any outcome that belongs to the event. For event A above (rolling an even number), the favourable outcomes are 2, 4 and 6. WORKED EXAMPLE 2 Identifying sample space, events and outcomes For the chance experiment of rolling a die: a. list the sample space b. list the events for: i. rolling a 4 ii. rolling an even number iii. rolling at least 5 iv. rolling at most 2 c. list the favourable outcomes for: i. {4, 5, 6} ii. not rolling 5 iii. rolling 3 or 4 iv. rolling 3 and 4. TOPIC 10 Probability 545  THINK WRITE a. The outcomes are the numbers 1 to 6. a. 𝜉 = {1, 2, 3, 4, 5, 6} b. i. This describes only one outcome. b. i. {4} ii. The possible even numbers are 2, 4 and 6. ii. {2, 4, 6} iii. ‘At least 5’ means 5 is the smallest. iii. {5, 6} iv. ‘At most 2’ means 2 is the largest. iv. {1, 2} c. i. The outcomes are shown inside the brackets. c. i. 4, 5, 6 ii. ‘Not 5’ means everything except 5. ii. 1, 2, 3, 4, 6 iii. The event is {3, 4}. iii. 3, 4 iv. There is no number that is both 3 and 4. iv. There are no favourable outcomes. 10.2.3 Theoretical probability eles-4879 When a coin is tossed, there are two possible outcomes, Heads or Tails. That is, 𝜉 = {H, T}. We assume the coin is unbiased, meaning each outcome is equally likely to occur. Since each outcome is equally likely, they must have the same probability. The total of all probabilities is 1, so the probability of an outcome is given by: 1 Pr(outcome) = number of outcomes 1 1 Pr(Heads) = and Pr(Tails) = 2 2 The probability of an event A is found by adding up all the probabilities of the favourable outcomes in event A. Theoretical probability When determining probabilities of equally likely outcomes, use the following: 1 Pr(outcome) = total number of outcomes number of favourable outcomes Pr(event A) = total number of outcomes WORKED EXAMPLE 3 Calculating theoretical probability A die is rolled and the number uppermost is noted. Determine the probability of each of the following events. a. A = {1} b. B = {odd numbers} c. C = {4 or 6} THINK WRITE There are six possible outcomes. 546 Jacaranda Maths Quest 9  1 a. A has one favourable outcome. a. Pr(A) = 6 3 b. B has three favourable outcomes: 1, 3 and 5. b. Pr(B) = 6 1 = 2 2 c. C has two favourable outcomes. c. Pr(C) = 6 1 = 3 Resources Resourceseses eWorkbook Topic 10 Workbook (worksheets, code puzzle and project) (ewbk-2010) Digital documents SkillSHEET Probability scale (doc-6307) SkillSHEET Understanding a deck of playing cards (doc-6308) SkillSHEET Listing the sample space (doc-6309) SkillSHEET Theoretical probability (doc-6310) Interactivities Individual pathway interactivity: Theoretical probability (int-4534) Probability scale (int-3824) Theoretical probability (int-6081) Exercise 10.2 Theoretical probability Individual pathways PRACTISE CONSOLIDATE MASTER 1, 3, 5, 8, 10, 14, 15, 19, 22 2, 4, 6, 9, 11, 16, 20, 23 7, 12, 13, 17, 18, 21, 24, 25 To answer questions online and to receive immediate corrective feedback and fully worked solutions for all questions, go to your learnON title at www.jacplus.com.au. Fluency 1. WE1 On the given probability scale, insert each of the following events at appropriate points. Indicate the chance of each event using one of the 0 0.5 1 following terms: certain, likely, unlikely, impossible. a. The school will have a lunch break on Friday. b. Australia will host two consecutive Olympic Games. 2. On the given probability scale, insert each of the following events at appropriate points. Indicate the chance of each event using one of the 0 0.5 1 following terms: certain, likely, unlikely, impossible. a. At least one student in a particular class will obtain an A for Mathematics. b. Australia will have a swimming team in the Commonwealth Games. TOPIC 10 Probability 547 3. On the given probability scale, insert each of the following events at appropriate points. Indicate the chance of each event using one of the 0 0.5 1 following terms: certain, likely, unlikely, impossible. a. Mathematics will be taught in secondary schools. b. In the future most cars will run without LPG or petrol. 4. On the given probability scale, insert each of the following events at appropriate points. Indicate the chance of each event using one of the following terms: certain, likely, unlikely, impossible. a. Winter will be cold. b. Bean seeds, when sown, will germinate. 0 0.5 1 5. WE2a For each chance experiment below, list the sample space. a. Rolling a die b. Tossing a coin c. Testing a light bulb to see whether it is defective or not d. Choosing a card from a normal deck and noting its colour e. Choosing a card from a normal deck and noting its suit 6. WE2b A normal six-sided die is rolled. List each of the following events. a. Rolling a number less than or equal to 3 b. Rolling an odd number c. Rolling an even number or 1 d. Not rolling a 1 or 2 e. Rolling at most a 4 f. Rolling at least a 5 7. WE2c A normal six-sided die is rolled. List the favourable outcomes for each of the following events. a. A = {3, 5} b. B = {1, 2} c. C = ‘rolling a number greater than 5’ d. D = ‘not rolling a 3 or a 4’ e. E = ‘rolling an odd number or a 2’ f. F = ‘rolling an odd number and a 2’ g. G = ‘rolling an odd number and a 3’ 8. A card is selected from a normal deck of 52 cards and its suit is noted. a. List the sample space. b. List each of the following events. i. Drawing a black card ii. Drawing a red card iii. Not drawing a heart iv. Drawing a black or a red card 9. Determine the number of outcomes there are for: a. rolling a die b. tossing a coin c. drawing a card from a standard deck d. drawing a card and noting its suit e. noting the remainder when a number is divided by 5. 548 Jacaranda Maths Quest 9 10. A card is drawn at random from a standard deck of 52 cards. Note: ‘At random’ means that every card has the same chance of being selected. Calculate the probability of selecting: a. an ace b. a king c. the 2 of spades d. a diamond. 11. WE3 A card is drawn at random from a deck of 52. Determine the probability of each event below. a. A = {5 of clubs} b. B = {black card} c. C = {5 of clubs or queen of diamonds} d. D = {hearts} e. E = {hearts or clubs} 12. A card is drawn at random from a deck of 52. Determine the probability of each event below. a. F = {hearts and 5} b. G = {hearts or 5} c. H = {aces or kings} d. I = {aces and kings} e. J = {not a 7} 13. A letter is chosen at random from the letters in the word PROBABILITY. Determine the probability that the letter is: a. B b. not B c. a vowel d. not a vowel. 14. The following coloured spinner is spun and the colour is noted. Determine the probability of each of the events given below. a. A = {blue} b. B = {orange} c. C = {orange or pink} d. D = {orange and pink} e. E = {not blue} Understanding 15. A bag contains four purple balls and two green balls. a. If a ball is drawn at random, then calculate the probability that it will be: i. purple ii. green. b. Design an experiment like the one in part a but where the probability of drawing a purple ball is 3 times that of drawing a green ball. 16. Design spinners (see question 14) using red, white and blue sections so that: a. each colour has the same probability of being spun b. red is twice as likely to be spun as either of the other two colours c. red is twice as likely to be spun as white and three times as likely to be spun as blue. TOPIC 10 Probability 549 17. A bag contains red, green and blue marbles. Calculate how many marbles there must be in the bag for the following to be true when a single marble is selected at random from the bag. a. Each colour is equally likely to be selected and there are at least six red marbles in the bag. b. Blue is twice as likely to be selected as the other colours and there are at least five green marbles in the bag. 1 1 1 c. There is a chance of selecting red, a chance of selecting green and a chance of selecting blue from 2 3 6 the bag when there is between 30 and 40 marbles in the bag. 18. a. A bag contains seven gold and three silver coins. If a coin is drawn at random from the bag, calculate the probability that it will be: i. gold ii. silver. b. After a gold coin is taken out of the bag, a second coin is then selected at random. Assuming the first coin was not returned to the bag, calculate the probability that the second coin will be: i. gold ii. silver. Reasoning 19. Do you think that the probability of tossing Heads is the same as the probability of tossing Tails if your friend tosses the coin? Suggest some reasons that it might not be. 20. If the following four probabilities were given to you, explain which two you would say were not correct. 0.725, −0.5, 0.005, 1.05 21. A coin is going to be tossed five times in a row. During the first four flips the coin comes up Heads each time. What is the probability that the coin will come up Heads again on the fifth flip? Justify your answer. Problem solving 22. Consider the spinner shown. Discuss whether the spinner has an equal chance of falling on each of the colours. 23. A box contains two coins. One is a double-headed coin, and the other is a normal coin with Heads on one side and Tails on the other. You draw one of the coins from a box and look at one of the sides. It is Heads. Determine the probability that the other side also shows Heads. 24. ‘Unders and Overs’ is a game played with two normal six-sided dice. The two dice are rolled, and the numbers uppermost added to give a total. Players bet on the outcome being ‘under 7’, ‘equal to 7’ or ‘over 7’. If you had to choose one of these outcomes, which would you choose? Explain why. 25. Justine and Mary have designed a new darts game for their Year 9 Fete Day. Instead of a circular dart board, their dart board is in the shape of two equilateral triangles. The inner triangle (bullseye) has a side length of 3 cm, while the outer triangle has side length 10 cm. Given that a player’s dart falls in one of the triangles, determine the 10 cm probability that it lands in the bullseye. Write your answer correct to 3 cm 2 decimal places. 550 Jacaranda Maths Quest 9  10.3 Experimental probability LEARNING INTENTION At the end of this subtopic you should be able to: determine the relative frequency of an outcome understand the difference between theoretical probability and experimental probability. 10.3.1 Relative frequency eles-4880 A die is rolled 12 times and the outcomes are recorded in the table shown. Outcome 1 2 3 4 5 6 Frequency 3 1 1 2 2 3 In this chance experiment there were 12 trials. The table shows that the number 1 was rolled 3 times out of 12. 3 1 So the relative frequency of 1 is 3 out of 12, or =. 12 4 As a decimal, the relative frequency of 1 is equal to 0.25. Relative frequency The relative frequency of an outcome is given by: frequency of an outcome Relative frequency = total number of trials As the number of trials becomes larger, the relative frequency of each outcome will become very close to the theoretical probability. WORKED EXAMPLE 4 Calculating relative frequency For the chance experiment of rolling a die, the following outcomes were noted. Outcome 1 2 3 4 5 6 Frequency 3 1 4 6 3 3 a. Calculate the number of trials. b. Identify how many threes were rolled. c. Calculate the relative frequency for each number written as a decimal. THINK WRITE a. Adding the frequencies a. 1 + 3 + 4 + 6 + 3 + 3 = 20 trials will give the number of trials. b. The frequency of 3 is 4. b. 4 threes were rolled. TOPIC 10 Probability 551  c. Add a relative frequency c. Outcome 1 2 3 4 5 6 row to the table and complete it. The relative Frequency 3 1 4 6 3 3 frequency is calculated 3 1 4 6 3 3 Relative = = = = = = by dividing the frequency 20 20 20 20 20 20 frequency of the outcome by the 0.15 0.05 0.2 0.3 0.15 0.15 total number of trials. 10.3.2 Experimental probability eles-4881 When it is not possible to calculate the theoretical probability of an outcome, carrying out simulations involving repeated trials can be used to determine the experimental probability. The relative frequency of an outcome is the experimental probability. Experimental probability The experimental probability of an outcome is given by: frequency of an outcome Experimental probability = total number of trials The spinner shown is not symmetrical, and the probability of each outcome cannot be determined theoretically. The experimental probability of each outcome can be found by using the spinner many times and recording the outcomes. As more trials are conducted, the experimental probability will become more accurate and closer to the true probability of each section. WORKED EXAMPLE 5 Calculating experimental probabilities The spinner shown above was spun 100 times and the following results were achieved. Outcome 1 2 3 4 Frequency 7 26 9 58 a. Calculate the number of trials. b. Calculate the experimental probability of each outcome. c. Calculate or recognise the sum of the four probabilities. THINK WRITE a. Adding the frequencies will determine the a. 7 + 26 + 9 + 58 = 100 trials number of trials. 552 Jacaranda Maths Quest 9  7 b. The experimental probability equals the relative b. Pr(1) = frequency. This is calculated by dividing the 100 frequency of the outcome by the total number = 0.07 of trials. 26 Pr(2) = 100 = 0.26 9 Pr(3) = 100 = 0.09 58 Pr(4) = 100 = 0.58 c. Add the probabilities (they should equal 1). c. 0.07 + 0.26 + 0.09 + 0.58 = 1 10.3.3 Expected number of results eles-4882 1 If we tossed a coin 100 times, we would expect there to be 50 Heads, since Pr (Heads) =. 2 Expected number of results The expected number of favourable outcomes from a series of trials is found from: Expected number = probability of outcome × number of trials The probability of an outcome can be the theoretical probability or an experimental probability. WORKED EXAMPLE 6 Expected number of outcomes Calculate the expected number of results in the following situations: a. The number of Tails after flipping a coin 250 times b. The number of times a one comes up after rolling a dice 120 times c. The number of times a royal card is picked from a deck that is reshuffled with the card replaced 650 times THINK WRITE 1 a. The probability of getting a Tail is Pr(Tails) =. a. Number of Tails = Pr(Tails) × 250 2 1 = × 250 2 = 125 times TOPIC 10 Probability 553  1 b. The probability of getting a one is Pr(one) =. b. Number of ones = Pr(one) × 120 6 1 = × 120 6 = 20 times c. A deck of cards has 52 cards and 12 of them are c. Number of royals = Pr(royal) × 650 royal cards. This means the probability of getting 3 12 3 = × 650 a royal is Pr (royal) = =. 13 52 13 = 150 times Digital technology Scientific calculators have a random function that allows the user to generate integers at random. This can be used to create a random set of data for probability or statistics questions. The button brings up the probability menu. The functions nPr, nCr and ! will be used in Year 10. Press across to select the RAND submenu and then randint( to generate random whole numbers. randint(1, 10) will generate a random integer between 1 and 10. The comma button is found by pressing the button and then the decimal button. COLLABORATIVE TASK Construct an irregular spinner using cardboard and a toothpick. By carrying out a number of trials, estimate the probability of each outcome. 554 Jacaranda Maths Quest 9  Resources Resourceseses eWorkbook Topic 10 Workbook (worksheets, code puzzle and project) (ewbk-2010) Interactivities Individual pathway interactivity: Experimental probability (int-4535) Experimental probability (int-3825) Exercise 10.3 Experimental probability Individual pathways PRACTISE CONSOLIDATE MASTER 1, 3, 8, 10, 11, 15, 17, 19, 22 2, 4, 6, 9, 12, 18, 20, 23, 24 5, 7, 13, 14, 16, 21, 25, 26, 27 To answer questions online and to receive immediate corrective feedback and fully worked solutions for all questions, go to your learnON title at www.jacplus.com.au. Fluency 1. WE4 Each of the two tables shown contains the results of a chance experiment (rolling a die). For each table, calculate: i. the number of trials held ii. the number of fives rolled iii. the relative frequency for each outcome, correct to 2 decimal places iv. the sum of the relative frequencies. Number 1 2 3 4 5 6 a. Frequency 3 1 5 2 4 1 Number 1 2 3 4 5 6 b. Frequency 52 38 45 49 40 46 2. A coin is tossed in two chance experiments. The outcomes are recorded in the tables shown. For each experiment, calculate: i. the relative frequency of both outcomes ii. the sum of the relative frequencies. Outcome H T a. Frequency 22 28 Outcome H T b. Frequency 31 19 3. WE5 An unbalanced die was rolled 200 times and the following outcomes were recorded. Number 1 2 3 4 5 6 Frequency 18 32 25 29 23 73 Using these results, calculate: a. Pr(6) b. Pr(odd number) c. Pr(at most 2) d. Pr(not 3). TOPIC 10 Probability 555 4. A box of matches claims on its cover to contain 100 matches. A survey of 200 boxes established the following results. Number of matches 95 96 97 98 99 100 101 102 103 104 Frequency 1 13 14 17 27 55 30 16 13 14 If you were to purchase a box of these matches, calculate the probability that: a. the box would contain 100 matches b. the box would contain at least 100 matches c. the box would contain more than 100 matches d. the box would contain no more than 100 matches. 5. A packet of chips is labelled as weighing 170 grams. This is not always the case and there will be some variation in the weight of each packet. A packet of chips is considered underweight if its weight is below 168 grams. Chips are made in batches of 1000 at a time. A sample of the weights from a particular batch are shown below. Weight (grams) 166 167 168 169 170 171 172 173 174 Frequency 2 3 13 17 27 18 9 5 1 Calculate the probability that: a. a packet of chips is its advertised weight b. a packet of chips is above its advertised weight c. a packet of chips is underweight. A batch of chips is rejected if more than 50 packets in a batch are classed as underweight. d. Explain if the batch that this sample of chips is taken from should be rejected. Understanding 6. Here is a series of statements based on experimental probability. If a statement is not reasonable, give a reason why. a. I tossed a coin five times and there were four Heads, so Pr(Heads) = 0.8. b. Sydney Roosters have won 1064 matches out of the 2045 that they have played, so Pr(Sydney will win their next game) = 0.52. c. Pr(the sun will rise tomorrow) = 1 d. At a factory, a test of 10 000 light globes showed that 7 were faulty. Therefore, Pr(faulty light globe) = 0.0007. e. In Sydney it rains an average of 143.7 days each year, so Pr(it will rain in Sydney on the 17th of next month) = 0.39. 7. At a birthday party, some cans of soft drink were put in a container of ice. There were 16 cans of Coke, 20 cans of Sprite, 13 cans of Fanta, 8 cans of Sunkist and 15 cans of Pepsi. If a can was picked at random, calculate the probability that it was: a. a can of Pepsi b. not a can of Fanta. 8. WE6 Calculate the expected number of Tails if a fair coin is tossed 400 times. 9. Calculate the expected number of threes if a fair die is rolled 120 times. 556 Jacaranda Maths Quest 9 10. MC In Tattslotto, six numbers are drawn from the numbers 1, 2, 3, … 45. The number of different combinations of six numbers is 8 145 060. If you buy one ticket, what is the probability that you will win the draw? 1 1 A. B. 8 145 060 45 45 1 C. D. 8 145 060 6 6 E. 8 145 060 11. MC A survey of high school students asked ‘Should Saturday be a normal school day?’ 350 students voted yes, and 450 voted no. From the following, recognise the probability that a student chosen at random said no. 7 9 7 9 1 A. B. C. D. E. 16 16 9 14 350 12. In a poll of 200 people, 110 supported party M, 60 supported party N and 30 were undecided. If a person is chosen at random from this group of people, calculate the probability that he or she: a. supports party M b. supports party N c. supports a party d. is not sure what party to support. 13. A random number is picked from N = {1, 2, 3, … 100}. Calculate the probability of picking a number that is: a. a multiple of 3 b. a multiple of 4 or 5 c. a multiple of 5 and 6. 14. The numbers 3, 5 and 6 are combined to form a three-digit number such that no digit may be repeated. a. i. Recognise how many numbers can be formed. ii. List them. b. Determine Pr(the number is odd). c. Determine Pr(the number is even). d. Determine Pr(the number is a multiple of 5). 15. MC In a batch of batteries, 2 out of every 10 in a large sample were faulty. At this rate, calculate how many batteries are expected to be faulty in a batch of 1500. A. 2 B. 150 C. 200 D. 300 E. 750 16. Svetlana, Sarah, Leonie and Trang are volleyball players. The probabilities that they will score a point on serve are 0.6, 0.4, 0.3 and 0.2 respectively. Calculate how many points on serve are expected from each player if they serve 10 times each. 17. MC A survey of the favourite leisure activity of 200 Year 9 students produced the following results. Activity Playing sport Fishing Watching TV Video games Surfing Number of students 58 26 28 38 50 The probability (given as a percentage) that a student selected at random from this group will have surfing as their favourite leisure activity is: A. 50% B. 100% C. 25% D. 0% E. 29% TOPIC 10 Probability 557 18. The numbers 1, 2 and 5 are combined to form a three-digit number, allowing for any digit to be repeated up to three times. a. Recognise how many different numbers can be formed. b. List the numbers. c. Determine Pr(the number is even). d. Determine Pr(the number is odd). e. Determine Pr(the number is a multiple of 3). Reasoning 19. John has a 12-sided die numbered 1 to 12 and Lisa has a 20-sided die numbered 1 to 20. They are playing a game where the first person to get the number 10 wins. They are rolling their dice individually. a. Calculate Pr(John gets a 10). b. Calculate Pr(Lisa gets a 10). c. Explain whether the game is fair. 20. At a supermarket checkout, the scanners have temporarily broken down and the cashiers must enter in the bar codes manually. One particular cashier overcharged 7 of the last 10 customers she served by entering the incorrect bar code. a. Based on the cashier’s record, determine the probability of making a mistake with the next customer. b. Explain if another customer should have any objections with being served by this cashier. 21. If you flip a coin six times, determine how many of the possible outcomes could include a Tail on the second toss. Problem solving 22. In a jar, there are 600 red balls, 400 green balls, and an unknown number of yellow balls. If the probability 1 of selecting a green ball is , determine how many yellow balls are in the jar. 5 23. In a jar there are an unknown number of balls, N, with 20 of them green. The other colours contained in the 1 1 1 jar are red, yellow and blue, with Pr(red or yellow) = , Pr(red or green) = and Pr(blue) =. 2 4 3 Determine the number of red, yellow and blue balls in the jar. 24. The biological sex of babies in a set of triplets is simulated by flipping three coins. If a coin lands Tails up, the baby is male. If a coin lands Heads up, the baby is female. In the simulation, the trial is repeated 40 times. The following results show the number of Heads obtained in each trial: 0, 3, 2, 1, 1, 0, 1, 2, 1, 0, 1, 0, 2, 0, 1, 0, 1, 2, 3, 2, 1, 3, 0, 2, 1, 2, 0, 3, 1, 3, 0, 1, 0, 1, 3, 2, 2, 1, 2, 1 a. Calculate the probability that exactly one of the babies in a set of triplets is female. b. Calculate the probability that more than one of the babies in the set of triplets is female. 25. Use your calculator to generate three random sets of numbers between 1 and 6 to simulate rolling a dice. The number of trials in each set will be: a. 10 b. 25 c. 50. Calculate the relative frequency of each outcome (1 to 6) and comment on what you notice about experimental and theoretical probability. 558 Jacaranda Maths Quest 9  26. A survey of the favourite foods of Year 9 students is recorded, with the following results. Meal Tally Hamburger 45 Fish and chips 31 Macaroni and cheese 30 Lamb souvlaki 25 BBQ pork ribs 21 Cornflakes 17 T-bone steak 14 Banana split 12 Corn-on-the-cob 9 Hot dogs 8 Garden salad 8 Veggie burger 7 Smoked salmon 6 Muesli 5 Fruit salad 3 a. Estimate the probability that macaroni and cheese is the favourite food of a randomly selected Year 9 student. b. Estimate the probability that a vegetarian dish is a randomly selected student’s favourite food. c. Estimate the probability that a beef dish is a randomly selected student’s favourite food. 27. A spinner has six sections of different sizes. Steven conducts an experiment and finds the following results: Section 1 2 3 4 5 6 Frequency 6 24 15 30 48 12 Determine the angle size of each section of the spinner using the results above. 10.4 Venn diagrams and two-way tables LEARNING INTENTION At the end of this subtopic you should be able to: identify the complement of an event A create and interpret Venn diagrams and two-way tables use a Venn diagram or two-way table to determine A ∩ B use a Venn diagram or two-way table to determine A ∪ B. 10.4.1 The complement of an event eles-4883 Suppose that a die is rolled. The sample space is given by: 𝜉 = {1, 2, 3, 4, 5, 6}. If A is the event ‘rolling an odd number’, then A = {1, 3, 5}. There is another event called ‘the complement of A′, or ‘not A′. This event contains all the outcomes that do not belong to A. It is given the symbol A′. In this case A′ = {2, 4, 6}. TOPIC 10 Probability 559  A and A′ can be shown on a Venn diagram. ξ ξ A A 1 1 3 5 3 5 6 6 4 4 2 2 ′ A is shaded. A (not A) is shaded. Complementary events For the event A, the complement is denoted A′ , and the two are related by the following: Pr(A) + Pr(A′ ) = 1 WORKED EXAMPLE 7 Determining the complement For the sample space 𝜉 = {1, 2, 3, 4, 5}, list the complement of each of the following events. a. A = {multiples of 3} b. B = {square numbers} c. C = {1, 2, 3, 5} THINK WRITE ′ a. The only multiple of 3 in the set is 3. Therefore A = {3}. A is every a. A′ = {1, 2, 4, 5} other element of the set. b. The only square numbers are 1 and 4. Therefore B = {1, 4}. B′ is b. B′ = {2, 3, 5} every other element of the set. c. C = {1, 2, 3, 5}. C′ is every other element of the set. c. C′ = {4} 10.4.2 Venn diagrams: the intersection of events eles-4884 A Venn diagram consists of a rectangle and one or more circles. A Venn diagram contains all possible outcomes in the sample space and will have the 𝜉 symbol in the top left corner. A Venn diagram is used to illustrate the relationship between sets of objects or numbers. All outcomes for a given event will be contained within a specific circle. Outcomes that belong to multiple events will be found in the overlapping region of two or more circles. The overlapping region of two circles is called the intersection of the two events and is represented using the ∩ symbol. 560 Jacaranda Maths Quest 9 ξ ξ ξ A B A B A B The circle on the left contains all The circle on the right contains all The overlap or intersection of the two outcomes in event A. outcomes in event B. circles contains the outcomes that are in event A ‘and’ in event B. This is denoted by A ∩ B. A Venn diagram for two events A and B has four distinct regions. ξ A ∩ B′ contains the outcomes in event A and not in event B. A ∩ B contains the outcomes in event A and in event B. A B A′ ∩ B contains the outcomes not in event A and in event B. A′ ∩ B′ contains the outcomes not in event A and not in event B. A ∩ B′ A∩B A′ ∩ B A′ ∩ B′ WORKED EXAMPLE 8 Setting up a Venn diagram In a class of 20 students, 5 study Art, 9 study Biology and 2 students study both. Let A = {students who study Art} and B = {students who study Biology}. a. Create a Venn diagram to represent this information. b. Identify the number of students represented by the following and state what these regions represent: i. A ∩ B ii. A ∩ B′ iii. A′ ∩ B iv. A′ ∩ B′ THINK WRITE/DRAW a. Draw a sample space with events A and B. a. ξ A B Place a 2 in the intersection of both circles since we know 2 students take both subjects. ξ A B Since 5 study Art and there are 2 already in the middle, place a 3 in the remaining section 3 2 7 of circle A. Since 9 study Biology and there are 2 already in the middle, place a seven in the remaining ξ section of circle B. A B The total number inside the three circles is 3 + 2 + 7 = 12. This means there must be 3 2 7 20 − 12 = 8 outside of the two circles. Place an 8 outside the circles, within the rectangle. 8 TOPIC 10 Probability 561  b. i. From the Venn diagram, A ∩ B = 2. These b. i. There are 2 students who study Art are the students in both A and B. and Biology. ξ A B 3 2 7 8 b. ii. From the Venn diagram, A ∩ B′ = 3. These b. ii. There are 3 students who study Art are the students in A and not in B. and not Biology, i.e. 3 students study ξ Art only. A B 3 2 7 8 b. iii. From the Venn diagram, A′ ∩ B = 7. These b. iii. There are 7 students who do not are the students not in A and in B. study Art and do study Biology, i.e. 7 students study Biology only. ξ A B 3 2 7 8 b. iv. From the Venn diagram, A′ ∩ B′ = 8. These b. iv. There are 8 students who do not are the students not in A and not in B. study Art and do not study Biology. ξ A B 3 2 7 8 10.4.3 Two-way tables eles-4885 The information in a Venn diagram can also be represented using a two-way table. The relationship between the two is shown below. Event B Event B' Total ξ A B Event A A∩B A ∩ B′ A A ∩ B′ A∩B A′ ∩ B Event A' A′ ∩ B A′ ∩ B′ A′ Total B B' A′ ∩ B′ 562 Jacaranda Maths Quest 9  WORKED EXAMPLE 9 Creating a two-way table In a class of 20 students, 5 study Art, 9 study Biology and 2 students study both. Create a two-way table to represent this information. THINK WRITE 1. Create an empty two-way table. Biology Not Biology Total Art Not Art Total 2. Fill the table in with the information Biology Not Biology Total provided in the question. 2 students study both subjects Art 2 5 5 in total take Art Not Art 9 in total take Biology Total 9 20 20 students in the class 3. Use the totals of the rows and columns Biology Not Biology Total to fill in the gaps in the table. 9 − 2 = 7 Biology and not Art Art 2 3 5 5 − 2 = 3 Art and not Biology Not Art 7 8 15 20 − 9 = 11 not Biology total Total 9 11 20 20 − 5 = 15 not Art total 4. The last value, not Biology and not Art, can be found from either of the following: 11 − 3 = 8 15 − 7 = 8 eles-4886 10.4.4 Number of outcomes If event A contains seven outcomes or members, this is written as n (A) = 7. So n (A ∩ B′) = 3 means that there are three outcomes that are in event A and not in event B. WORKED EXAMPLE 10 Determining the number of outcomes in an event For the Venn diagram shown, write down the number of outcomes in each of the following. a. M b. M′ c. M ∩ N d. M ∩ N′ e. M′ ∩ N′ ξ N M 11 15 6 4 TOPIC 10 Probability 563  THINK WRITE/DRAW a. Identify the regions showing M and add the a. ξ outcomes. N M 11 15 6 4 n (M) = 6 + 11 = 17 b. Identify the regions showing M′ and add the b. ξ outcomes. N M 11 15 6 4 n (M′) = 4 + 15 = 19 c. M ∩ N means ‘M and N’. Identify the region. c. ξ N M 11 15 6 4 n (M ∩ N) = 11 d. M ∩ N′ means ‘M and not N’. Identify the region. d. ξ N M 11 15 6 4 n (M ∩ N′) = 6 e. M′ ∩ N′ means ‘not M and not N’. Identify the e. ξ region. N M 11 15 6 4 n (M′ ∩ N′) = 4 564 Jacaranda Maths Quest 9 WORKED EXAMPLE 11 Using a Venn diagram to create a two-way table Show the information from the Venn diagram on a two-way table. ξ B A 2 7 3 5 THINK WRITE ′ 1. Draw a 2 × 2 table and add the labels A, A , A A′ B and B′. B B′ 2. There are 7 elements in A and B. A A′ There are 3 elements in A and ‘not B’. There are 2 elements in ‘not A’ and B. B 7 2 There are 5 elements in ‘not A’ and ‘not B’. B′ 3 5 3. Add in a column and a row to show the totals. A A′ Total B 7 2 9 B′ 3 5 8 Total 10 7 17 WORKED EXAMPLE 12 Using a two-way table to create a Venn diagram Show the information from the two-way table on a Venn diagram. Left-handed Right-handed Blue eyes 7 20 Not blue eyes 17 48 THINK DRAW Draw a Venn diagram that includes a sample ξ space and events L for left-handedness and B B for blue eyes. (Right-handedness = L′) L n(L ∩ B) = 7 7 20 n(L ∩ B′ ) = 17 17 n(L′ ∩ B) = 20 n(L′ ∩ B′ ) = 48 48 TOPIC 10 Probability 565  WORKED EXAMPLE 13 Probability from a two-way table In a class of 30 students, 15 swim for exercise and 20 run for exercise and 5 participate in neither activity. a. Create a two-way table to represent this information. b. Calculate the probability that a randomly chosen student from this class does running and swimming for exercise. THINK WRITE a. 1. Create an empty two-way table. a. Swim Not swim Total Run Not run Total 2. Fill the table in with the information provided Swim Not swim Total in the question. Run 20 5 students do neither activity. 15 in total swim. Not run 5 20 in total run. Total 15 30 30 students are in the class. 3. Use the totals of the rows and columns to fill Swim Not swim Total in the gaps in the table. Run 10 10 20 Not run 5 5 10 Total 15 15 30 number in run ∩ swim b. The probability a student runs and swims is b. Pr (run ∩ swim) = given by: total in the class number in run ∩ swim 10 Pr (run ∩ swim) = = total in the class 30 1 = 3 10.4.5 Venn diagrams: the union of events eles-4887 The intersection of two events (A ∩ B) is all outcomes in event A ‘and’ in event B. The union of two events (A ∪ B) is all outcomes in events A ‘or’ in event B. ξ ξ A B A B The intersection of the two circles contains the Everything contained within the two circles is an outcomes that are in event A ‘and’ in event B. outcome that is in event A‘or’ in event B. This is denoted by A ∩ B. This is denoted by A ∪ B. 566 Jacaranda Maths Quest 9 WORKED EXAMPLE 14 Calculating the union of two events Use the Venn diagram shown to calculate the value of the ξ following: A B a. n(A) b. Pr(B) c. n(A ∪ B) 29 16 31 d. Pr(A ∩ B) 44 THINK WRITE a. Identify the number of outcomes in the a. n(A) = 29 + 16 = 45 A circle. favourable outcomes in B b. Identify the number of outcomes in the b. Pr(B) = B circle. total number of outcomes number of favourable outcomes in B n(B) Pr(B) = = total number of outcomes total number of outcomes 16 + 31 = 29 + 16 + 31 + 44 47 = 120 c. Identify the number of outcomes in either c. n(A ∪ B) = 29 + 16 + 31 of the two circles. = 76 number of favourable outcomes in A ∩ B d. Identify the number of outcomes in the d. Pr(A ∩ B) = intersection of the two circles. total number of outcomes n(A ∩ B) favourable outcomes in A ∩ B = Pr(A ∩ B) = total number of outcomes total number of outcomes 16 = 29 + 16 + 31 + 44 16 = 120 2 = 15 WORKED EXAMPLE 15 Completing a Venn diagram a. Place the elements of the following sets of numbers in their correct position in a single Venn diagram. 𝜉 = {Number 1 to 20 inclusive} A = {Multiples of 3 from 1 to 20 inclusive} B = {Multiples of 2 from 1 to 20 inclusive} b. Use this Venn diagram to determine the following: i. A ∩ B ii. A ∪ B iii. A ∩ B′ iv. A′ ∪ B′ TOPIC 10 Probability 567  THINK WRITE/DRAW a. Write out the numbers in each event a. ξ A and B: A ={3, 6, 9, 12, 15, 18 } A B 2, 4, 8 A ={2, 4, 6, 8, 10, 12, 14, 16, 18, 20} 3, 9, 15 6, 12, 18 10, 14, 16, Identify the numbers that appear in both 20 sets. In this case it is 6, 12 and 18. These numbers will be placed in the overlap of the two circles for A and B. 1, 5, 7, 11, 13, 17, 19 All numbers not in A or B are placed outside the two circles. After placing the numbers in the Venn diagram, check that all numbers from 1 to 20 are written down. b. i. A ∩ B are the numbers in A ‘and’ in B. b. i. A ∩ B = {6,12,18} ii. A ∪ B are the numbers in A ‘or’ in B. ii. A ∪ B = {2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20} iii. A ∩ B′ are the number in A and not in B. iii. A ∩ B′ = {3, 9, 15} Refer to the four sections of the Venn diagram to locate this region. { } ′ 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, ′ iv. A′ ∪ B′ is any number that is not in A ‘or’ iv. A ∪ B = 14, 15, 16, 17, 19, 20 not in B. This ends up being any number not in A ∩ B. WORKED EXAMPLE 16 Calculating the probability of the union of two events In a class of 24 students, 11 students play basketball, 7 play tennis, and 4 play both sports. a. Show the information on a Venn diagram. b. If one student is selected at random, then calculate the probability that: i. the student plays basketball ii. the student plays tennis or basketball iii. the student plays tennis or basketball but not both. THINK WRITE/DRAW a. 1. Draw a sample space with events a. ξ B and T. T B 568 Jacaranda Maths Quest 9  2. n(B ∩ T) = 4 ξ n(B ∩ T′) = 11 − 4 = 7 T n(T ∩ B′) = 7 − 4 = 3 B 3 So far, 14 students out of 24 have been 4 placed. 7 n(B′ ∩ T′) = 24 − 14 = 10 10 number of students who play basketball b. i. Identify the number of students who b. i. Pr(B) = play basketball. total number of students n(B) ξ = 24 T 11 B = 4 3 24 7 10 number of favourable outcomes Pr(B) = total number of outcomes n(T ∪ B) ii. Identify the number of students who ii. Pr(T ∪ B) = play tennis or basketball. 24 14 ξ = 24 T 7 = B 3 12 4 7 10 iii. Identify the number of students who iii. n(B ∩ T′) + n(B′ ∩ T) = 3 + 7 play tennis or basketball but not both. = 10 ξ Pr(tennis or basketball but not both) T 10 = B 24 4 3 5 = 7 12 10 DISCUSSION How will you remember the difference between when one event and another occurs and when one event or another occurs? TOPIC 10 Probability 569  Resources Resourceseses eWorkbook Topic 10 Workbook (worksheets, code puzzle and project) (ewbk-2010) Digital documents SkillSHEET Determining complementary events (doc-6311) SkillSHEET Calculating the probability of a complementary event (doc-6312) Video eLesson Venn diagrams (eles-1934) Interactivities Individual pathway interactivity: Venn diagrams and two-way tables (int-4536) Venn diagrams (int-3828) Two-way tables (int-6082) Exercise 10.4 Venn diagrams and two-way tables Individual pathways PRACTISE CONSOLIDATE MASTER 1, 4, 5, 10, 13, 16, 17, 20, 24 2, 6, 7, 9, 11, 14, 18, 21, 25 3, 8, 12, 15, 19, 22, 23, 26, 27 To answer questions online and to receive immediate corrective feedback and fully worked solutions for all questions, go to your learnON title at www.jacplus.com.au. Fluency 1. WE7 For the sample space 𝜉 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, list the complement of each of the following events. a. A = {evens} b. B = {multiples of 5} c. C = {sqaures} d. D = {numbers less than 8} 2. If 𝜉 = {11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, list the complement of each of the following events. a. A = {multiples of 3} b. B = {numbers less than 20} c. C = {prime numbers} d. D = {odd numbers or numbers greater than 16} 3. If 𝜉 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, list the complement of each of the following events. a. A = {multiples of 4} b. B = {primes} c. C = {even and less than 13} d. D = {even or greater than 13} 4. WE10 For the Venn diagram shown, write down the number of outcomes in: ξ a. 𝜉 b. S T c. T d. T ∩ S S 7 e. T ∩ S′ f. S′ ∩ T′. 6 5 9 5. WE11 Show the information from question 4 on a two-way table. 570 Jacaranda Maths Quest 9  6. WE12 Show the information from this two-way table on a Venn diagram. S S′ V 21 7 V′ 2 10 7. For each of the following Venn diagrams, use set notation to write the name of the region coloured in: i. blue ii. pink. a. b. c. ξ ξ ξ A B A B W 8. WE9, 13 The membership of a tennis club consists of 55 men and 45 women. There are 27 left-handed people, including 15 men. a. Show the information on a two-way table. b. Show the information on a Venn diagram. c. If one member is chosen at random, calculate the probability that the person is: i. right-handed ii. a right-handed man iii. a left-handed woman. 9. WE14 Using the information given in the Venn diagram, if one outcome is chosen at random, determine: a. Pr(L) b. Pr(L′) c. Pr(L ∩ M) d. Pr(L ∩ M′). ξ M L 7 5 3 10 10. WE15 Place the elements of the following sets of numbers in their correct position in a single Venn diagram. 𝜉 = {numbers between 1 to 10 inclusive} A = {odd numbers from 1 to 10} B = {squared numbers between 1 to 10 inclusive} 11. Place the elements of the following sets of numbers in their correct position in a single Venn diagram. 𝜉 = {numbers between 1 to 25 inclusive} A = {multiples of 3 from 1 to 25} B = {numbers that are odd or over 17 from 1 to 25 inclusive} TOPIC 10 Probability 571 12. Place the elements of the following sets of numbers in their correct pos