Contents

1. D4 Probability • D2 Contents D4.2 The probability scale • D2 D4.3 Calculating probability • D2 D4.1 The language of probability D4.4 Probability diagrams • D2 D4.5 Experimental probability • D2

2. Probability is a measurement of the chance or likelihood of an event happening. The language of probability Describe the chance of drawing a red marble. even Chance متساوي الفرصة Unlikely غير مرجح Certain مؤكد Impossible مستحيل Likely مرجح

3. Meeting with King Henry VIII The next baby born being a boy Getting a number > 2 when roll a fair dice A square having four right angles A day of the week starting with a T Less likely More likely The chance of an event happening can be shown on a probability scale. The probability scale impossible unlikely even chance certain likely

4. bag b bag b A game is played with marbles in a bag. One of the following bags is chosen for the game. The teacher then pulls a marble at random from the chosen bag: Fair games bag a bag c If a red marble is pulled out of the bag, the girls get a point. If a blue marble is pulled out of the bag, the boys get a point. Which would be the fair bag to use?

5. A game is fair if all the players have an equal chance of winning. Fair games Which of the following games are fair? A dice is thrown. If it lands on a prime number team A gets a point, if it doesn’t team B gets a point. There are three prime numbers (2, 3 and 5) and three non-prime numbers (1, 4 and 6). Yes, this game is fair.

6. Nine cards numbered 1 to 9 are used and a card is drawn at random. If a multiple of 3 is drawn team A gets a point. If a square number is drawn team B gets a point. If any other number is drawn team C gets a point. Fair games There are three multiples of 3 (3, 6 and 9). There are three square numbers (1, 4 and 9). There are four numbers that are neither square nor multiples of 3 (2, 5, 7 and 8). No, this game is not fair. Team C is more likely to win.

7. 5 1 4 2 3 A spinner has five equal sectors numbered 1 to 5. The spinner is spun many times. If the spinner stops on an even number team A gets 3 points. If the spinner stops on an odd number team B gets 2 points. Fair games Suppose the spinner is spun 50 times. We would expect the spinner to stop on an even number 20 times and on an odd number 30 times. Team A would score 20 × 3 points = 60 points Team B would score 30 × 2 points = 60 points Yes, this game is fair.

8. bag c Choose a blue counter and win a prize! Bags of counters bag a bag b bag c You are only allowed to choose one counter at random from one of the bags. Which of the bags is most likely to win a prize?

9. Meeting with King Henry VIII The next baby born being a boy Getting a number > 2 when roll a fair dice A square having four right angles A day of the week starting with a T Less likely More likely The chance of an event happening can be shown on a probability scale. The probability scale impossible unlikely even chance certain likely

10. 0 ½ 1 We measure probability on a scale from 0 to 1. If an event is impossible or has no probability of occurring then it has a probability of 0. The probability scale If an event is certain it has a probability of 1. This can be shown on the probability scale as: impossible even chance certain Probabilities are written as fractions, decimal and, less often, as percentages between 0 and 1.

11. The probability scale

12. D4 Probability D4.1 The language of probability • D2 Contents D4.2 The probability scale • D2 • D2 D4.3 Calculating probability D4.4 Probability diagrams • D2 D4.5 Experimental probability • D2

13. Higher or lower

14. 1 1 1 1 1 1 6 6 6 6 6 6 1 Since each number on the dice is equally likely the probability of getting any one of the numbers is 1 divided by 6 or . 6 When you roll a fair dice you are equally likely to get one of six possible outcomes: Listing possible outcomes

15. 1) A coin landing tails up? 3) Drawing a seven of hearts from a pack of 52 cards? 1 P(7 of ) = 52 1 1 1 7 2 4 2) This spinner stopping on the red section? 4) A baby being born on a Friday? What is the probability of the following events? Calculating probability P(tails) = P(Friday) = P(red) =

16. Number of successful outcomes Probability of an event = Total number of possible outcomes 3 10 If the outcomes of an event are equally likely then we can calculate the probability using the formula: Calculating probability For example, a bag contains 1 yellow, 3 green, 4 blue and 2 red marbles. What is the probability of pulling a green marble from the bag without looking? P(green) = or 30% or 0.3

17. 4 1 2 1 1 = 8 8 2 4 8 = This spinner has 8 equal divisions: What is the probability of the spinner landing on • a red sector? • a blue sector? • a green sector? Calculating probability a) P(red) = b) P(blue) = c) P(green) =

18. 3 1 2 1 1 6 6 3 2 6 = = • A fair dice is thrown. What is the probability of getting • a 2? • a multiple of 3? • an odd number? • a prime number? • a number bigger than 6? • an integer? Calculating probability a) P(2) = b) P(a multiple of 3) = c) P(an odd number) =

19. 3 1 6 = 0 6 2 6 Don’t write 6 • A fair dice is thrown. What is the probability of getting • a 2? • a multiple of 3? • an odd number? • a prime number? • a number bigger than 6? • an integer? Calculating probability d) P(a prime number) = e) P(a number bigger than 6) = 0 f) P(an integer) = = 1

20. Calculating probabilities Answer these questions giving each answer as a fraction or 0 or 1.

21. 1 3 4 4 The following spinner is spun once: The probability of an event not occurring What is the probability of it landing on the yellow sector? P(yellow) = What is the probability of it not landing on the yellow sector? P(not yellow) = If the probability of an event occurring is p then the probability of it not occurring is 1 – p.

22. The probability of pulling a picture card out of a full deck of cards is . What is the probability of not pulling out a picture card? 3 10 3 P(not a picture card) = 1 – = 13 13 13 The probability of a factory component being faulty is 0.03. What is the probability of a randomly chosen component not being faulty? The probability of an event not occurring P(not faulty) = 1 – 0.03 = 0.97

23. Event A B C D 3 2 5 5 11 9 20 20 The following table shows the probabilities of 4 events. For each one work out the probability of the event not occurring. The probability of an event not occurring Probability of the event occurring Probability of the event not occurring 0.77 0.23 8% 92%

24. 10 are cola bottles, are fried eggs, 20 are hearts, the rest are teddies. 1 4 5 6 45 3 = 60 4 There are 60 sweets in a bag. The probability of an event not occurring What is the probability that a sweet chosen at random from the bag is: P(not a cola bottle) = a) Not a cola bottle P(not a teddy) = b) Not a teddy

25. For example, a game is played with the following cards: 2 1 1 1 1 and P(sun) = 3 3 3 3 3 + = If two outcomes are mutually exclusive then their probabilities can be added together to find their combined probability. Adding mutually exclusive outcomes What is the probability that a card is a moon or a sun? P(moon) = Drawing a moon and drawing a sun are mutually exclusive outcomes so, P(moon or sun) = P(moon) + P(sun) =

26. 1 1 1 1 5 and P(star) = 3 3 9 3 3 1 3 + 3 – 1 + – = = 9 9 Adding mutually exclusive outcomes What is the probability that a card is yellow or a star? P(yellow card) = Drawing a yellow card and drawing a star are not mutually exclusive outcomes because a card could be yellow and a star. P (yellow card or star) cannot be found simply by adding. We have to subtract the probability of getting a yellow star. P(yellow card or star) =

27. P(blue) = 0.15 P(yellow) = 0.4 P(green) = 0.35 The sum of all mutually exclusive outcomes is 1. For example, a bag contains red counters, blue counters, yellow counters and green counters. The sum of all mutually exclusive outcomes What is the probability of drawing a red counter from the bag? P(blue or yellow or green) = 0.15 + 0.4 + 0.35 = 0.9 P(red) = 1 – 0.9 = 0.1

28. 1) We can list them systematically. TH and HT are separate equally likely outcomes. Two coins are thrown. What is the probability of getting two heads? Before we can work out the probability of getting two heads we need to work out the total number of equally likely outcomes. Finding all possible outcomes of two events There are three ways to do this: Using H for heads and T for tails, the possible outcomes are: TT, TH, HT, HH.

29. Second coin H T First coin H T 1 4 2) We can use a two-way table. Finding all possible outcomes of two events HH HT TH TT From the table we see that there are four possible outcomes one of which is two heads so, P(HH) =

30. H H T H T 1 T 4 3) We can use a probability tree diagram. Outcomes Second coin Finding all possible outcomes of two events HH First coin HT TH TT Again we see that there are four possible outcomes so, P(HH) =

31. A red dice and a blue dice are thrown and their scores are added together. Finding the sample space What is the probability of getting a total of 8 from both dice? There are several ways to get a total of 8 by adding the scores from two dice. We could get a 2 and a 6, a 3 and a 5, a 4 and a 4, a 5 and a 3, or a 6 and a 2. To find the set of all possible outcomes, the sample space, we can use a two-way table.

32. 8 5 8 8 8 8 36 + From the sample space we can see that there are 36 possible outcomes when two dice are thrown. 2 3 4 5 6 7 Finding the sample space 3 4 5 6 7 8 4 5 6 7 8 9 5 6 7 8 9 10 Five of these have a total of 8. 6 7 8 9 10 11 P(8) = 7 8 9 10 11 12

33. D4 Probability D4.1 The language of probability • D2 Contents D4.2 The probability scale • D2 D4.3 Calculating probability • D2 D4.5 Experimental probability D4.4 Probability diagrams • D2 • D2

34. From this we can estimate the probability of someone being left-handed as or 0.087. 87 1000 Suppose 1000 people were asked whether they were left- or right-handed. Of the 1000 people asked 87 said that they were left-handed. Estimating probabilities based on data If we repeated the survey with a different sample the results would probably be slightly different. The more people we asked, however, the more accurate our estimate of the probability would be.

35. Relative frequency is calculated using the formula: Number of successful trials Relative frequency = Total number of trials 65 13 Relative frequency = = 100 20 The probability of an event based on data from an experiment or survey is called the relative frequency. Relative frequency For example, Ben wants to estimate the probability that a piece of toast will land butter-side-down. He drops a piece of toast 100 times and observes that it lands butter-side-down 65 times.

36. Number Frequency Relative frequency 30 32 42 27 38 31 1 31 200 200 200 200 200 200 2 27 3 38 4 30 5 42 6 32 Sita wants to know if her dice is fair. She throws it 200 times and records her results in a table: Relative frequency = 0.155 = 0.135 = 0.190 = 0.150 = 0.210 = 0.160 Is the dice fair?

37. Experimental probability

38. The theoretical probability of getting a 5 is . 1 6 1 So, expected frequency = × 300 = 6 The theoretical probability of an event is its calculated probability based on equally likely outcomes. If the theoretical probability of an event can be calculated, then when we do an experiment we can work out the expected frequency. Expected frequency Expected frequency = theoretical probability × number of trials If you rolled a dice 300 times, how many times would you expect to get a 5? 50

39. 1 2 2 3 If you tossed a coin 250 times how many times would you expect to get a tail? Expected frequency Expected frequency = × 250 = 125 If you rolled a fair dice 150 times how many times would you expect to a number greater than 2? Expected frequency = × 150 = 100

40. Spinners experiment

41. Worksheet ( 1 ) Zayed althani school Math department * Write the sample space ( all possible results ) when rolling a fair dice اكتب فضاء العينة ( مجموعة جميع النواتج الممكنة ) عند إلقاء حجر نرد منتظم Mohamad badawi : hamadaa_math@yahoo.com Use the words : impossible , unlikely , even chance , likely , certain to describe the following events : استخدم المصطلحات : مستحيل ، غير مرجح ، متساوي الفرصة ، مرجح ، مؤكد لتصف الأحداث التالية : 1) The upper face is a number greater than 5. ………………. ظهور عدد اكبر من 5 على الوجه العلوي 2) The upper face is a prime number. ………………. ظهور عدد أولي على الوجه العلوي * You toss 2 coins together ألقيت قطعتي نقود معاً 1) Complete the table to show all possible results . أكمل الجدول لتبين جميع النواتج الممكنة Use the words : impossible , unlikely , even chance , likely , certain to describe the following events : استخدم المصطلحات : مستحيل ، غير مرجح ، متساوي الفرصة ، مرجح ، مؤكد لتصف الأحداث التالية : 1) You will get 2 heads ………………. ستحصل على صورتين 2) At least one head ………………. ستظهر صورة واحدة على الأقل 3) You will get one tail exactly………………. ستظهر الكتابة مرة واحدة بالضبط

42. Worksheet ( 2 ) Zayed althani school Math department 2 cards were randomly drawn from a deck of 52 cards تم سحب ورقتين من علبة لعب الورق ( الشدة ) التي تحوي 52 ورقة Mohamad badawi : hamadaa_math@yahoo.com 1) Complete the table to show all possible results . أكمل الجدول لتبين جميع النواتج الممكنة Use the words : impossible , unlikely , even chance , likely , certain to describe the following events : استخدم المصطلحات : مستحيل ، غير مرجح ، متساوي الفرصة ، مرجح ، مؤكد لتصف الأحداث التالية : 1) The 2 cards are of the same color. ………………. البطاقتان من نفس اللون 2) The 2 cards are spades. ………………. البطاقتان من نوع البستوني 3) One of the cards was green. ………………. احد البطاقتين خضراء : Spade بستوني 4) The 2 cards are either red or black or red and black………………. : Clubs سباتي البطافتان إما حمراوتان أو سوداوتان أو حمراء وسوداء : Diamond ديناري 5) At least one of the 2 cards wasn’t a picture. ………………. : Heart كبه ( قلب ) على الأقل احدهما ليست صورة