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17 ES Ch apter N AL PA G Statistics and Probability Probability FI In this chapter, we introduce probability, which deals with how likely it is that something will happen. This is an area of mathematics with many diverse applications. The study of probability began in seventeenth-century France, when the two great French mathematicians Blaise Pascal (1623–1662) and Pierre de Fermat (1601–1665) corresponded about problems from games of chance. Problems such as these continued to influence the early development of the subject. Nowadays probability is used in areas ranging from weather forecasting and insurance, where it is used to calculate risk factors and premiums, to predicting the risks and benefits of new medical treatments. 42 8 Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400 17A An introduction to probability Most people would agree with the following statements: • It is certain that the sun will rise tomorrow. • If I toss a coin, getting a head and getting a tail are equally likely. Everyday language PA G ES • There is no chance of finding a plant that speaks English. N AL Using everyday language to discuss probabilities can cause problems, because people do not always agree on the interpretation of words such as ‘likely’, ‘probable’ and ‘certain’. Consider these two examples: 1 Two farmers are discussing the prospects of getting a good wheat crop this year. Farmer Bill says, ‘I don’t think it is likely to rain for the next two weeks. I’m not going to plant wheat yet.’ Farmer Tony says, ‘I reckon you’re wrong. I’m certain we’ll have rain. It can’t go on the way it has. I’m getting the tractor out tomorrow.’ FI 2 Alanna is captain of the Platypus Netball Team. They are going to play the Echidnas, whose captain is Maria. Each captain says to her team before the match, ‘I think we’ll probably win. Just follow the plans we’ve practised all week.’ Alanna and Maria cannot both be right! It would be futile to try to assign a probability that the Echidnas (or the Platypuses) are going to win on the basis of what the captains told their teams. On the other hand, there are many situations in which it would be useful to be able to measure how likely, or unlikely, it is that an event will occur. We can do this in mathematics by using the idea of probability, which we define as a number between 0 and 1 that we assign to any event we are interested in. A probability of 1 represents an event that is ‘certain’ or ‘guaranteed to happen’. A probability of 0 represents an event that we would describe as ‘impossible’ or one that ‘cannot possibly occur’. 1 An event that has a probability is as likely to occur as not to occur. 2 C h apt e r 1 7 P r o babi l it y Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400 429 1 7 A A n i n t r o d u c t i o n t o p r o b a b i l i t y An event that has a probability close to 0 is unlikely to occur. An event that has a probability close to 1 is likely to occur. 1 2 0 Impossible 1 Certain Using these ideas, let us look at the three statements at the start of this section and express them in the new language of probability. Statement in terms of probability It is certain that the sun will rise tomorrow. The probability that the sun will rise tomorrow is 1. If I toss a coin, getting a head and getting a tail are equally likely. If I toss a coin, the probability of getting a head is 1 and the probability of getting a tail is 1. 2 2 The probability of finding a plant that speaks English is 0. Example 1 PA G There is no chance of finding a plant that speaks English. ES Original statement FI N AL A TV game show contestant is shown three closed doors and told that there is a prize behind only one of the doors. If the contestant opens one of the doors, what is his probability of winning the prize? Solution From the point of view of the quiz contestant the prize is equally likely to be behind each of 1 the doors. So the probability of the contestant winning the prize is . 3 430 I C E - E M M at h em at ic s y e a r 7 Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400 Exercise 17A Example 1 1 Complete each of the following probability statements. a It is certain that an iceblock will melt in the sun in Sydney. The probability that an iceblock will melt in the sun in Sydney is _______ b There is no chance of finding a dog that speaks German. The probability of finding a dog that speaks German is _______ ES c If there are three red discs and three blue discs in a bag and you take one out without looking in the bag, you are equally likely to get a red disc or a blue disc. If there are three red discs and three blue discs in a bag and you take one out without looking in the bag, the probability of getting a red disc is _______ and the probability of getting a blue disc is _______ d If it is Thursday today, tomorrow will be Friday. If it is Thursday today, the probability that it will be Friday tomorrow is _______ G e If today is the 31st of January, there is no chance that tomorrow will be the 1st of May. If today is the 31st of January, the probability that tomorrow is the 1st of May is _______ N AL 17 B PA 2 Make up some statements for which there is a corresponding probability statement 1 involving the probabilities 0, 1 or . 2 Experiments and counting A standard die is a cube with each face marked with a number from 1 to 6 or with marks as shown in the diagram. FI Suppose we roll a standard die. Since the outcomes 1, 2, ..., 6, are equally likely (assuming that the die is fair; that is, it is not ‘loaded’), the probability of getting 1 1 a 1 is , the probability of getting a 2 is , and so on. 6 6 The total probability is 1 Question: If we roll a die, what is the chance of getting 1 or 2 or 3 or 4 or 5 or 6? Answer: It is certain that we will get one of the numbers 1, 2, 3, 4, 5, 6, so we say that the total probability is 1. C h apt e r 1 7 P r o babi l it y Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400 431 1 7 B E x pe r i m e n t s a n d c o u n t i n g Sample space Rolling a die and recording the face that shows up is an example of doing an experiment. The numbers 1, 2, ..., 6, are called the outcomes of this experiment. The complete set of possible outcomes for any experiment is called the sample space of that experiment. For example, we can write down the sample space for the experiment of rolling a die as: 1 2 3 4 5 6 1 In this case, each outcome is equally likely and has probability . 6 Events ES An event is something that happens. In everyday life, we speak of sporting events, or we may say that the school concert was a memorable event. We use the word ‘event’ in probability in a similar way. G For example, suppose that we roll a die and we are interested in getting a prime number. In this case ‘the number is prime’ is the event that interests us. Some of the outcomes will give rise to this event. For instance, if the outcome is 2, then the event ‘the number is prime’ has occurred. We say that the outcome 2 is favourable to the event ‘the number is prime’. If the outcome is 4, then the event ‘the number is prime’ does not occur. The outcome 4 is not favourable to the event. Of the six outcomes, three – 2, 3 and 5 – are favourable to the event ‘the number is prime’. Three outcomes – 1, 4 and 6 – are not favourable to the event ‘the number is prime’. Example 2 PA In many situations ‘success’ means favourable to the event and ‘failure’ means not favourable to the event. Hassan rolls a die. Hassan is hoping for a perfect square number. N AL a What is the sample space? b What are the outcomes that are favourable to the event ‘the result is a perfect square’? Solution FI a The sample space is: 1 2 3 4 5 6 b The only perfect squares in the sample space are 1 and 4, so the outcomes favourable to the event are: 1 4 Probability of an event Suppose I roll a die. A natural question to ask is: What is the probability that I get a 1 or a 6? To make sense of this, we have to decide how the question is to be interpreted. There are six possible outcomes: 1 2 3 4 5 6 432 I C E - E M M at h em at ic s y e a r 7 Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400 1 7 B E x pe r i m e n t s a n d c o u n t i n g All are equally likely. Of these, only two are favourable to the event ‘the result is 1 or 6’. They are: 1 6 2 1 = because all of the outcomes 1, 2, 3, 4, 5 6 3 and 6 are equally likely. In general, the situation is as follows. We say that the probability of getting a 1 or a 6 is Probability of an event For an experiment in which all of the outcomes are equally likely: number of outcomes favourable to that event total number of outcomes ES Probability of an event = Here is a good way of setting out the solution to a question about the probability of an event. G Example 3 PA I roll a die. What is the probability that the result is a perfect square? Solution The sample space of all possible outcomes is: 1 2 3 4 5 6 N AL The outcomes that are favourable to the event ‘the result is a perfect square’ can be circled, as shown. 1 2 3 4 5 6 FI Probability that the result is a perfect square = number of favourable outcomes total number of outcomes 2 6 1 = 3 = Notation From now on, we will often write ‘P’ instead of ‘probability of’. Thus we write the previous result as: P(perfect square) = 1 3 C h apt e r 1 7 P r o babi l it y Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400 433 1 7 B E x pe r i m e n t s a n d c o u n t i n g Example 4 I roll a die. What is the probability of getting a 2 or a 3 or a 4 or a 5? Solution The sample space, with the favourable outcomes circled, is: 1 2 3 4 5 6 number of favourable outcomes total number of outcomes 4 = 6 2 = 3 The word ‘not’ PA P(2 or 3 or 4 or 5) = 1 − P(1 or 6). Why is this so? G Class discussion ES P(2 or 3 or 4 or 5) = Sometimes we need to recognise the word ‘not’ hidden in the question. The next example gives a simple illustration of what we mean by this. N AL Example 5 Joe is playing a board game. If he tosses a 1 with the die, he goes to jail. What is the probability that he does not go to jail? Solution FI This is the same as the probability of not getting a 1, which means that Joe gets a 2, 3, 4, 5 or 6. So P (not a 1) = P (2, 3, 4, 5 or 6) 5 = 6 Alternatively, we can consider the probability of getting anything except a 1. Then P (not a 1) = 1 − P (1) 1 =1− 6 5 = , as before. 6 434 I C E - E M M at h em at ic s y e a r 7 Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400 1 7 B E x pe r i m e n t s a n d c o u n t i n g The words ‘random’ and ‘randomly’ In probability, we frequently hear expressions such as ‘chosen randomly’ or ‘chosen at random’, as in the following situations: • A classroom contains 23 students. A teacher comes into the room and chooses a student at random to answer a question about history. What does this mean? It means that the teacher chose the student as if she knew nothing at all about the students. Another way of interpreting this is to imagine that the teacher had her eyes closed and had no idea who was in the class when she chose a student. • In a TV quiz show, a prize wheel is spun. The contestant is asked to guess which number the wheel will stop at. ES The results of the spins are random and the contestant has no idea where the wheel will stop. Example 6 PA Solution G In a pick-a-box show, there are five closed boxes, each containing one snooker ball. Two of the boxes each contain yellow balls and the others contain a green ball, a black ball and a red ball, respectively. The contestant will win a prize if she chooses a yellow ball. She chooses a box at random and opens it. What is the probability that she wins a prize? Here is the sample space with the favourable outcomes circled. They are equally likely, because she chooses at random. N AL Y Y G B R 2 P(prize) = 5 Example 7 A school has 1200 students. The table below gives information about whether or not each student plays a musical instrument. Girls Plays a musical instrument 325 450 Does not play an instrument 225 200 FI Boys Each student has a school number between 1 and 1200. Each student’s number is written on a card and the 1200 cards are placed in a hat. One card is chosen randomly from the hat. What is the probability that the number pulled out belongs to: a b c d a boy? a girl? a student who does not play an instrument? a boy who plays a musical instrument? C h apt e r 1 7 P r o babi l it y Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400 435 1 7 B E x pe r i m e n t s a n d c o u n t i n g Solution b Number of students = 1200 Number of girls = 450 + 200 = 650 650 P(girl’s number) = 1200 13 = 24 a Number of student = 1200 Number of boys = 325 + 225 = 550 550 P (boy’s number) = 1200 11 = 24 G ES c Number of students = 1200 Number who do not play a musical instrument = 225 + 200 = 425 425 P(student does not play a musical instrument) = 1200 17 = 48 d Number of students = 1200 N AL PA Number of boys who play a musical instrument = 325 325 P(boy who plays a musical instrument) = 1200 13 = 48 Exercise 17B Example 2 1 A bag contains 10 marbles numbered 1 to 10. A marble is taken out. a Give the sample space for this experiment. FI b Write down the outcomes favourable to the event ‘a marble with an odd number is taken out’. 2 A die is rolled. Write down the outcomes favourable to each given event. a An even number is rolled. b An odd number is rolled. c A number divisible by 3 is rolled. d A number greater than 1 is rolled. 436 3 b Write down the outcomes favourable to the event ‘the number obtained is less than 4’. 1 4 2 3 aFor the spinner shown opposite, write down the sample space of the experiment ‘spinning the spinner’. 5 I C E - E M M at h em at ic s y e a r 7 Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400 Example 3, 4 1 7 B E x pe r i m e n t s a n d c o u n t i n g 4 A die is rolled. Work out the probability that: a an even number is rolled b an odd number is rolled c a number divisible by 3 is rolled d a number greater than 1 is rolled 5 A bag contains 10 marbles numbered 1 to 10. If a marble is taken out at random, what is the probability of getting: a a 3? b an even number? c a number greater than 2? d a number divisible by 3? e a number greater than 1? 6 The spinner shown opposite is spun. Write down the probability of: 3 b obtaining an odd number 1 4 ES 2 a obtaining a 4 5 PA G 7 The letter tiles for the word MELBOURNE are placed in a box. One piece is withdrawn at random. What is the probability of obtaining: a an M? c a vowel? d a consonant? 8 A die is rolled. What is the probability of: N AL Example 5 b an E? a obtaining a 3? b not obtaining a 3? c obtaining a number divisible by 3? d obtaining a number not divisible by 3? 9 A bag contains three red marbles numbered 1 to 3, five green marbles numbered 4 to 8, and two yellow marbles numbered 9 and 10. A single marble is withdrawn at random. Find the probability that: b the marble is green c the marble is yellow d the number is odd e the number is greater than 6 f the number is green and even FI a the marble is numbered 3 10 The letters of the alphabet are written on flashcards and put into a bag. A card is drawn out at random. Find the probability that the letter is: a D b not W, X, Y or Z c a vowel d in the word ‘PERTH’ e in the word ‘CANBERRA’ C h apt e r 1 7 P r o babi l it y Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400 437 1 7 B E x pe r i m e n t s a n d c o u n t i n g Example 6 11 There are three green apples and two red apples in a bowl. Georgia is blindfolded and randomly chooses one of the apples. What is the probability that she chooses a green apple? 12 A bag of sweets contains 10 red ones, nine green ones, six yellow ones and five blue ones. They are all the same size and shape. You reach in and take one out at random. Find the probability that the sweet you pick is: a blue b green or yellow d purple e not yellow or blue c not red 13 Cards with numbers 1–100 written on them are placed in a box. One card is randomly chosen. What is the probability of obtaining a number divisible by 5? Normal jelly beans Double-flavoured jelly beans Green Red 250 400 175 75 ES 14 A bowl contains green and red normal jelly beans and green and red double-flavoured jelly beans. The numbers of the different colours and types are given in the table below. G Example 7 A jelly bean is randomly taken out of the bowl. Find the probability that: b it is a green jelly bean PA a it is a double-flavoured jelly bean c it is a green normal jelly bean e it is a red jelly bean d it is a red double-flavoured jelly bean 15 In a traffic survey taken during a 30-minute period, the number of people in each passing car was noted, and the results tabulated as follows. 1 2 3 4 5 Number of cars 60 50 40 10 5 N AL Number of people in car What is the probability that: a there was only one person in a car during this period? b there was more than one person in a car during this period? c there were fewer than four people in a car during this period? FI d there were five people in a car during this period? 16 A door prize is to be awarded randomly at the end of a concert. The numbers of people at the concert in different age groups are given in the table below. Age group 0−5 6−11 12−18 19−30 30−40 40+ Number of people in age group 10 150 350 420 125 85 What is the probability of the prize winner being: 438 a in the 0–5 age group? b in the 19–30 age group? c aged 40 or less? d aged between 12 and 30? e older than 5? f older than 40? I C E - E M M at h em at ic s y e a r 7 Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400 Review exercise 1 There are eight marbles, numbered 1 to 8, in a bowl. A marble is randomly taken out. What is the probability of getting: a an 8? b an even number? c a number less than 6? d a number divisible by 3? 2 Twelve cards are put into a hat. Five of the cards are red and are numbered 1 to 5, and the other seven cards are blue and are numbered 6 to 12. A card is taken randomly from the hat. Find the probability of taking out: b a card with an even number c a card with a prime number d a red card ES a a blue card e a card with a number greater than 13 a the P? b a B? G 3 The letter tile pieces for the letters of the word ‘PROBABILITY’ are put into a box. One piece is chosen at random. What is the probability of choosing: c a vowel? d a consonant? PA 4 Seven red marbles numbered 1 to 7 and six blue marbles numbered 8 to 13 are placed in a box. A marble is taken randomly from the box. Find the probability of taking out: a a red marble b a blue marble c a marble with a prime number d a marble with a multiple of 3 e a red marble with an even number f a blue marble with an odd number N AL g a red marble with a number less than 6 on it 5 A large bowl contains two types of lollies: chocolates and toffees. The chocolates and toffees are manufactured by the Yummy Sweet Company and the ACE Confectionary Company. The number of chocolates and toffees in the bowl of the two different brands are given in the table below. ACE Confectionary Company 250 170 300 140 FI Chocolates Toffees Yummy Sweet Company A lolly is randomly taken out of the bowl. Find the probability that: a it is a chocolate manufactured by the Yummy Sweet Company b it is a toffee manufactured by the Ace Confectionary Company c it is a chocolate d it is a toffee 6 The 11 letter tiles for the letters of the word ‘PROSPECTIVE’ are put into a box. One piece is chosen at random. What is the probability of choosing: a the I? b a P? c an E? d a vowel? C h apt e r 1 7 P r o babi l it y Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400 439 7 Each number from 1 to 500 is written on a card and put into a box. A card is withdrawn at random. What is the probability of obtaining a card with: a a number less than 100? b a number divisible by 10? c a number divisible by 5? Challenge exercise ES 1 In a raffle, 50 tickets are sold. The tickets are numbered from 1 to 50. They are placed in a hat and one is drawn out at random to win a prize. Find the probability that the number on the winning ticket: b is not 28 c is even d is less than or equal to 20 e is an even number less than 20 f contains the digit 7 h contains the digit 2 at least once PA g does not contain the digit 9 G a is 43 i contains the digit 2 only once j does not contain the digit 2 2 Suppose that the raffle tickets in Question 1 are coloured: those numbered 1 to 25 are green, the tickets numbered 26 to 40 are yellow and the rest are blue. Find the probability that the ticket that wins the prize: a is blue b is not green d is both green and blue e is even-numbered and blue f is prime and yellow N AL c is green or blue 3 Thomas, Leslie, Anthony, Tracie and Kim play a game in which there are three equal prizes. No player can win more than one prize. a List the 10 ways the prizes can be allocated. b What is the probability that Leslie wins a prize? FI c What is the probability that Leslie does not win a prize? 4 Find the probability that a randomly chosen three-digit number is divisible by: a three b seven 5 A bag initially contains 15 red balls and 8 blue balls. Some of the red balls are removed. 1 A ball is then drawn out at random. The probability that it is a red ball is . How many 3 red balls were removed? 6 Describe how you can use a coin to decide between three menu items with equal probability. 7 The six faces of a die are labelled with the numbers −3, −2, −1, 0, 1, and 2. The die is rolled twice. What is the probability that the product of the two numbers is negative (less than zero)? 440 I C E - E M M at h e matic s y e a r 7 Final pages • Cambridge University Press © Brown et al, 2017 • ISBN 978-1-108-40124-1 • Ph 03 8671 1400