$ and det A = 14, find the possible values of p. 1. If A =! # Use your graph to answer parts (i) (iii) below, Working:

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1 & 2 p 3 1. If A =! # $ and det A = 14, find the possible values of p. % 4 p p" Use your graph to answer parts (i) (iii) below, (i) Find an estimate for the median score. (ii) Candidates who scored less than 35 were required to retake the examination. How many candidates had to retake? Answer: (iii) The highest-scoring 15% of candidates were awarded a distinction. Find the mark above which a distinction was awarded. (Total 16 marks)... (Total 4 marks) 3. A box contains 35 red discs and 5 black discs. A disc is selected at random and its colour noted. The disc is then replaced in the box. In eight such selections, what is the probability that a black disc is selected 2. One thousand candidates sit an examination. The distribution of marks is shown in the following grouped frequency table. (i) exactly once? Marks Number of candidates (ii) at least once? The process of selecting and replacing is carried out 400 times. Copy and complete the following table, which presents the above data as a cumulative frequency distribution. What is the expected number of black discs that would be drawn? (Total 8 marks) Mark Number of candidates Draw a cumulative frequency graph of the distribution, using a scale of 1 cm for 100 candidates on the vertical axis and 1 cm for 10 marks on the horizontal axis. (5) 1 2

2 4. For the events A and B, p(a) = 0.6, p(b) = 0.8 and p(a B) = 1. Find p(a B); p( A B). 6. The mean of the population x 1, x 2,..., x 25 is m. Given that x i = 300 and 25 i= 1 2 ( x m) = 625, find i the value of m; the standard deviation of the population. 25 i= 1 Answers: (Total 4 marks) Answers: (Total 4 marks) 5. At a conference of 100 mathematicians there are 72 men and 28 women. The men have a mean height of 1.79 m and the women have a mean height of 1.62 m. Find the mean height of the 100 mathematicians. 7. The graph shows a normal curve for the random variable X, with mean µ and standard deviation σ. y A 0 12 x Answer:... (Total 4 marks) It is known that p (X 12) = 0.1. The shaded region A is the region under the curve where x 12. Write down the area of the shaded region A. (1) 3 4

3 It is also known that p (X 8) = 0.1. Find the value of µ, explaining your method in full. Show that σ = 1.56 to an accuracy of three significant figures. (d) Find p (X 11). (5) (5) (5) (Total 16 marks) 9. In a survey, 100 students were asked do you prefer to watch television or play sport? Of the 46 boys in the survey, 33 said they would choose sport, while 29 girls made this choice. Television Boys Girls Total Sport Total A fair coin is tossed eight times. Calculate the probability of obtaining exactly 4 heads; the probability of obtaining exactly 3 heads; (1) By completing this table or otherwise, find the probability that a student selected at random prefers to watch television; a student prefers to watch television, given that the student is a boy. the probability of obtaining 3, 4 or 5 heads. Answers: (Total 4 marks) 10. A supermarket records the amount of money d spent by customers in their store during a busy period. The results are as follows: Money in $ (d) Number of customers (n)

4 Find an estimate for the mean amount of money spent by the customers, giving your answer to the nearest dollar ($). (i) Write down the values of a and b. (ii) Find the probability that the lifespan of an insect of this species is Copy and complete the following cumulative frequency table and use it to draw a cumulative frequency graph. Use a scale of 2 cm to represent $20 on the horizontal axis, and 2 cm to represent 20 customers on the vertical axis. Money in $ (d) <20 <40 <60 <80 < 100 < 120 < 140 (5) more than 55 hours; between 55 and 60 hours. (1) Number of customers (n) % of the insects die after t hours. The time t (minutes), spent by customers in the store may be represented by the equation (i) Represent this information on a standard normal curve diagram, similar to the one given in part, indicating clearly the area representing 90%. (i) 2 t = 2d Use this equation and your answer to part to estimate the mean time in minutes spent by customers in the store. (ii) Find the value of t. (Total 10 marks) (ii) Use the equation and the cumulative frequency graph to estimate the number of customers who spent more than 37 minutes in the store. (5) (Total 15 marks) 12. Two ordinary, 6-sided dice are rolled and the total score is noted. Complete the tree diagram by entering probabilities and listing outcomes. 6 Outcomes The lifespan of a particular species of insect is normally distributed with a mean of 57 hours and a standard deviation of 4.4 hours. The probability that the lifespan of an insect of this species lies between 55 and 60 hours is represented by the shaded area in the following diagram. This diagram represents the standard normal curve not not 6 a 0 b... not

5 14. The following Venn diagram shows a sample space U and events A and B. Find the probability of getting one or more sixes. U A B n(u) = 36, n(a) = 11, n(b) = 6 and n(a B) = 21. Answer:... (Total 4 marks) On the diagram, shade the region (A B). Find (i) n(a B); (ii) P(A B). Explain why events A and B are not mutually exclusive. 13. The table shows the scores of competitors in a competition. Score Number of competitors with this score k 3 Answers: (i)... The mean score is 34. Find the value of k. (ii) (Total 4 marks) Answer:... (Total 4 marks) 9 10

6 15. A survey is carried out to find the waiting times for 100 customers at a supermarket. waiting time (seconds) number of customers Calculate an estimate for the mean of the waiting times, by using an appropriate approximation to represent each interval. Construct a cumulative frequency table for these data. Use the cumulative frequency table to draw, on graph paper, a cumulative frequency graph, using a scale of 1 cm per 20 seconds waiting time for the horizontal axis and 1 cm per 10 customers for the vertical axis. (1) (d) Use the cumulative frequency graph to find estimates for the median and the lower and upper quartiles. (Total 10 marks)

7 13 14

8 15 16

9 ( 3 ) 17 18

10 19 20

11 21 22

12 ( 1 ) 23 24

13 ( 2 ) 25 26

14 k s ) If a person is selected at random from this group of 200, find the probability that this person is 1 (i) (ii) an unemployed female; a male, given that the person is employed Calculate an estimate for the mean of the waiting times, by using an appropriate approximation to represent each interval. Construct a cumulative frequency table for these data. (1) Answers: (i)... (ii)... (Total 4 marks) (d) Use the cumulative frequency table to draw, on graph paper, a cumulative frequency graph, using a scale of 1 cm per 20 seconds waiting time for the horizontal axis and 1 cm per 10 customers for the vertical axis. Use the cumulative frequency graph to find estimates for the median and the lower and upper quartiles. (Total 10 marks) 18. The following diagram represents the lengths, in cm, of 80 plants grown in a laboratory frequency In a survey of 200 people, 90 of whom were female, it was found that 60 people were unemployed, including 20 males. 5 Using this information, complete the table below. Males Females Totals length (cm) Unemployed Employed How many plants have lengths in cm between Totals 200 (i) 50 and 60? (ii) 70 and 90? 27 28

15 Calculate estimates for the mean and the standard deviation of the lengths of the plants. 19. Given the following frequency distribution, find the median; Explain what feature of the diagram suggests that the median is different from the mean. (1) the mean. Number (x) Frequency (f ) (d) The following is an extract from the cumulative frequency table. length in cm cumulative less than frequency Answers:..... Use the information in the table to estimate the median. Give your answer to two significant figures. (Total 10 marks)... (Total 4 marks) 20. A bag contains 10 red balls, 10 green balls and 6 white balls. Two balls are drawn at random from the bag without replacement. What is the probability that they are of different colours? Answer:... (Total 4 marks) 29 30

16 21. The table below represents the weights, W, in grams, of 80 packets of roasted peanuts. A cumulative frequency graph of the distribution is shown below, with a scale 2 cm for 10 packets on the vertical axis and 2 cm for 5 grams on the horizontal axis. Weight (W) 80 < W < W < W < W < W < W < W 115 Number of packets Use the midpoint of each interval to find an estimate for the standard deviation of the weights Copy and complete the following cumulative frequency table for the above data. Weight (W) W 85 W 90 W 95 W 100 W 105 W 110 W 115 Number of packets (1) 50 Number of packets Weight (grams) Use the graph to estimate (i) (ii) the median; the upper quartile (that is, the third quartile). Give your answers to the nearest gram

17 (d) Let W, W,..., W be the individual weights of the packets, and let W be their mean. What is the value of the sum ( W 1 W ) + ( W2 W ) + ( W3 W ) ( W79 W ) + ( W80 W )? 23. The following Venn diagram shows the universal set U and the sets A and B. U B A (e) One of the 80 packets is selected at random. Given that its weight satisfies 85 < W 110, find the probability that its weight is greater than 100 grams. (Total 14 marks) Shade the area in the diagram which represents the set B A'. 22. Intelligence Quotient (IQ) in a certain population is normally distributed with a mean of 100 and a standard deviation of 15. What percentage of the population has an IQ between 90 and 125? If two persons are chosen at random from the population, what is the probability that both have an IQ greater than 125? n(u) = 100, n(a) = 30, n(b) = 50, n(a B) = 65. Find n(b A ). An element is selected at random from U. What is the probability that this element is in B A? The mean IQ of a random group of 25 persons suffering from a certain brain disorder was found to be Is this sufficient evidence, at the 0.05 level of significance, that people suffering from the disorder have, on average, a lower IQ than the entire population? State your null hypothesis and your alternative hypothesis, and explain your reasoning. (Total 9 marks) Answers: (Total 4 marks) 33 34

18 24. The events B and C are dependent, where C is the event a student takes Chemistry, and B is the event a student takes Biology. It is known that 25. The speeds in km h 1 of cars passing a point on a highway are recorded in the following table. P(C) = 0.4, P(B C) = 0.6, P(B C%) = 0.5. Speed v Number of cars Complete the following tree diagram. v 60 0 Chemistry Biology 60 < v < v C B 80 < v < v < v B! 110 < v C! B 120 < v < v v > B! Calculate an estimate of the mean speed of the cars. Calculate the probability that a student takes Biology. Given that a student takes Biology, what is the probability that the student takes Chemistry? Answers: (Total 4 marks) The following table gives some of the cumulative frequencies for the information above. Speed v Cumulative frequency v 60 0 v 70 7 v v v 100 a v v 120 b v v

19 27. From January to September, the mean number of car accidents per month was 630. From October to December, the mean was 810 accidents per month. (i) Write down the values of a and b. What was the mean number of car accidents per month for the whole year? (ii) On graph paper, construct a cumulative frequency curve to represent this information. Use a scale of 1 cm for 10 km h 1 on the horizontal axis and a scale of 1 cm for 20 cars on the vertical axis. (5) Use your graph to determine (i) (ii) the percentage of cars travelling at a speed in excess of 105 km h 1; the speed which is exceeded by 15% of the cars. (Total 11 marks) Answer: Bags of cement are labelled 25 kg. The bags are filled by machine and the actual weights are normally distributed with mean 25.7 kg and standard deviation 0.50 kg. What is the probability a bag selected at random will weigh less than 25.0 kg? 28. A box contains 22 red apples and 3 green apples. Three apples are selected at random, one after the other, without replacement. The first two apples are green. What is the probability that the third apple is red? What is the probability that exactly two of the three apples are red? In order to reduce the number of underweight bags (bags weighing less than 25 kg) to 2.5% of the total, the mean is increased without changing the standard deviation. Show that the increased mean is 26.0 kg. It is decided to purchase a more accurate machine for filling the bags. The requirements for this machine are that only 2.5% of bags be under 25 kg and that only 2.5% of bags be over 26 kg. Calculate the mean and standard deviation that satisfy these requirements. The cost of the new machine is $5000. Cement sells for $0.80 per kg. Answers: (d) Compared to the cost of operating with a 26 kg mean, how many bags must be filled in order to recover the cost of the new equipment? (Total 11 marks) 37 38

20 29. A taxi company has 200 taxi cabs. The cumulative frequency curve below shows the fares in dollars ($) taken by the cabs on a particular morning. 200 Use the curve to estimate (i) the median fare; (ii) the number of cabs in which the fare taken is $35 or less The company charges 55 cents per kilometre for distance travelled. There are no other charges. Use the curve to answer the following. On that morning, 40% of the cabs travel less than a km. Find the value of a What percentage of the cabs travel more than 90 km on that morning? (Total 10 marks) Number of cabs Two fair dice are thrown and the number showing on each is noted. The sum of these two numbers is S. Find the probability that S is less than 8; at least one die shows a 3; 40 at least one die shows a 3, given that S is less than 8. (Total 7 marks) The mass of packets of a breakfast cereal is normally distributed with a mean of 750 g and standard deviation of 25 g Fares ($) Find the probability that a packet chosen at random has mass (i) less than 740 g; (ii) at least 780 g; (iii) between 740 g and 780 g. (5) 39 40

21 Two packets are chosen at random. What is the probability that both packets have a mass which is less than 740 g? The mass of 70% of the packets is more than x grams. Find the value of x. (Total 9 marks) 33. Three positive integers a, b, and c, where a < b < c, are such that their median is 11, their mean is 9 and their range is 10. Find the value of a For events A and B, the probabilities are P (A) =, P (B) = Calculate the value of P (A B) if 6 P (A B) = ; 11 events A and B are independent. Answer:... Answers:

22 34. In a suburb of a large city, 100 houses were sold in a three-month period. The following cumulative frequency table shows the distribution of selling prices (in thousands of dollars). Selling price P ($1000) Total number of houses P 100 P 200 P 300 P 400 P A standard doorway in Tallopia is designed so that 99% of adults have a space of at least 17 cm over their heads when going through a doorway. Find the height of a standard doorway in Tallopia. Give your answer to the nearest cm. (Total 7 marks) Represent this information on a cumulative frequency curve, using a scale of 1 cm to represent $50000 on the horizontal axis and 1 cm to represent 5 houses on the vertical axis. Use your curve to find the interquartile range. The information above is represented in the following frequency distribution. Selling price P ($1000) Number of houses 0 < P < P < P < P < P a b Find the value of a and of b. (d) (e) Use mid-interval values to calculate an estimate for the mean selling price. Houses which sell for more than $ are described as De Luxe. (i) (ii) Use your graph to estimate the number of De Luxe houses sold. Give your answer to the nearest integer. Two De Luxe houses are selected at random. Find the probability that both have a selling price of more than $ (Total 15 marks) 35. In a country called Tallopia, the height of adults is normally distributed with a mean of cm and a standard deviation of 9.5 cm. What percentage of adults in Tallopia have a height greater than 197 cm? 43 44

23 36. The number of hours of sleep of 21 students are shown in the frequency table below. Hours of sleep Number of students Consider events A, B such that P (A) 0, P (A) 1, P (B) 0, and P (B) 1. In each of the situations,, below state whether A and B are mutually exclusive (M); independent (I); neither (N). P(A B) = P(A) P(A B) = 0 P(A B) = P(A) Find the median; the lower quartile; the interquartile range. Answers: Answers:

24 38. A family of functions is given by A student is selected at random. f (x) = x 2 + 3x + k, where k {1, 2, 3, 4, 5, 6, 7}. (i) Calculate the probability that he studies both economics and history. One of these functions is chosen at random. Calculate the probability that the curve of this function crosses the x-axis. (ii) Given that he studies economics, calculate the probability that he does not study history. A group of three students is selected at random from the school. Answer:... (i) (ii) Calculate the probability that none of these students studies economics. Calculate the probability that at least one of these students studies economics. (5) (Total 12 marks) 39. In a school of 88 boys, 32 study economics (E), 28 study history (H) and 39 do not study either subject. This information is represented in the following Venn diagram. E (32) H (28) U (88) 40. A company manufactures television sets. They claim that the lifetime of a set is normally distributed with a mean of 80 months and standard deviation of 8 months. What proportion of television sets break down in less than 72 months? (i) Calculate the proportion of sets which have a lifetime between 72 months and 90 months. (ii) Illustrate this proportion by appropriate shading in a sketch of a normal distribution curve. (5) a b c 39 If a set breaks down in less than x months, the company replace it free of charge. They replace 4% of the sets. Find the value of x. (Total 10 marks) Calculate the values a, b, c

25 Cumulative frequency 41. A student measured the diameters of 80 snail shells. His results are shown in the following cumulative frequency graph. The lower quartile (LQ) is 14 mm and is marked clearly on the graph. 42. A painter has 12 tins of paint. Seven tins are red and five tins are yellow. Two tins are chosen at random. Calculate the probability that both tins are the same colour LQ = Diameter (mm) Answer:... On the graph, mark clearly in the same way and write down the value of (i) the median; (ii) the upper quartile. Write down the interquartile range. 43. It is claimed that the masses of a population of lions are normally distributed with a mean mass of 310 kg and a standard deviation of 30 kg. Calculate the probability that a lion selected at random will have a mass of 350 kg or more. The probability that the mass of a lion lies between a and b is 0.95, where a and b are symmetric about the mean. Find the value of a and of b. (Total 5 marks) Answer:

26 Use the cumulative frequency curve to complete the frequency table below. 44. The cumulative frequency curve below shows the marks obtained in an examination by a group of 200 students Mark (x) 0 x < x < x < x < x < 100 Number of students Forty percent of the students fail. Find the pass mark Number of students Answer: Mark obtained 51 52

27 45. Let A and B be events such that P(A) = 2 1, P(B) = 4 3 and P(A B) = The table below shows the marks gained in a test by a group of students. Calculate P(A B). Calculate P(A(B). Are the events A and B independent? Give a reason for your answer. Mark Number of students 5 10 p 6 2 The median is 3 and the mode is 2. Find the two possible values of p. Answers: Answer: Dumisani is a student at IB World College. 7 The probability that he will be woken by his alarm clock is. 8 1 If he is woken by his alarm clock the probability he will be late for school is. 4 3 If he is not woken by his alarm clock the probability he will be late for school is. 5 Let W be the event Dumisani is woken by his alarm clock. Let L be the event Dumisani is late for school

28 Copy and complete the tree diagram below. L 48. The cumulative frequency curve below shows the heights of 120 basketball players in centimetres W 100 L! L W! 60 Number of players 50 L! Calculate the probability that Dumisani will be late for school Given that Dumisani is late for school what is the probability that he was woken by his alarm clock? (Total 11 marks) Height in centimetres 55 56

29 Use the curve to estimate the median height; the interquartile range. The arrow is spun. It cannot land on the lines between the sectors. Let A, B, C and S be the events defined by A: Arrow lands in sector A B: Arrow lands in sector B C: Arrow lands in sector C S: Arrow lands in a shaded region. Find Answers: P(B); P(S); P(A(S) The following diagram shows a circle divided into three sectors A, B and C. The angles at the centre of the circle are 90, 120 and 150. Sectors A and B are shaded as shown. Answers:.. C A B 57 58

30 50. Let a, b, c and d be integers such that a < b, b < c and c = d. The mode of these four numbers is 11. The range of these four numbers is 8. The mean of these four numbers is 8. Calculate the value of each of the integers a, b, c, d. 51. A packet of seeds contains 40% red seeds and 60% yellow seeds. The probability that a red seed grows is 0.9, and that a yellow seed grows is 0.8. A seed is chosen at random from the packet. Complete the probability tree diagram below. 0.9 Grows 0.4 Red Does not grow Grows Answers: a =..., b =... c =..., d =... Yellow Does not grow (i) Calculate the probability that the chosen seed is red and grows. (ii) (iii) Calculate the probability that the chosen seed grows. Given that the seed grows, calculate the probability that it is red. (7) (Total 10 marks) 59 60

31 52. Reaction times of human beings are normally distributed with a mean of 0.76 seconds and a standard deviation of 0.06 seconds. 53. A test marked out of 100 is written by 800 students. The cumulative frequency graph for the marks is given below. The graph below is that of the standard normal curve. The shaded area represents the probability that the reaction time of a person chosen at random is between 0.70 and 0.79 seconds a 0 b (i) Write down the value of a and of b. Number of candidates (ii) Calculate the probability that the reaction time of a person chosen at random is 200 greater than 0.70 seconds; between 0.70 and 0.79 seconds. (6) Three percent (3%) of the population have a reaction time less than c seconds. Mark (i) Represent this information on a diagram similar to the one above. Indicate clearly the area representing 3%. (ii) Find c. (Total 10 marks) 61 62

32 Write down the number of students who scored 40 marks or less on the test. The middle 50% of test results lie between marks a and b, where a < b. Find a and b A factory makes calculators. Over a long period, 2% of them are found to be faulty. A random sample of 100 calculators is tested. Write down the expected number of faulty calculators in the sample. Find the probability that three calculators are faulty. Find the probability that more than one calculator is faulty

33 55. The speeds of cars at a certain point on a straight road are normally distributed with mean µ and standard deviation σ. 15% of the cars travelled at speeds greater than 90 km h 1 and 12% of them at speeds less than 40 km h 1. Find µ and σ Bag A contains 2 red balls and 3 green balls. Two balls are chosen at random from the bag without replacement. Let X denote the number of red balls chosen. The following table shows the probability distribution for X X P(X = x) 3 10 Calculate E(X), the mean number of red balls chosen. Bag B contains 4 red balls and 2 green balls. Two balls are chosen at random from bag B. (i) Draw a tree diagram to represent the above information, including the probability of each event. (ii) Hence find the probability distribution for Y, where Y is the number of red balls chosen. A standard die with six faces is rolled. If a 1 or 6 is obtained, two balls are chosen from bag A, otherwise two balls are chosen from bag B. (d) Calculate the probability that two red balls are chosen. Given that two red balls are obtained, find the conditional probability that a 1 or 6 was rolled on the die. (Total 19 marks) (8) (5) 65 66

34 57. Two unbiased 6-sided dice are rolled, a red one and a black one. Let E and F be the events E : the same number appears on both dice; F : the sum of the numbers is 10. Find P(E); P(F); P(E F). 58. The 45 students in a class each recorded the number of whole minutes, x, spent doing experiments on Monday. The results are x = Find the mean number of minutes the students spent doing experiments on Monday. Two new students joined the class and reported that they spent 37 minutes and 30 minutes respectively. Calculate the new mean including these two students. Answers: Answers: The table below shows the subjects studied by 210 students at a college. Year 1 Year 2 Totals History Science Art Totals

35 A student from the college is selected at random. Let A be the event the student studies Art. Let B be the event the student is in Year 2. (i) Find P(A). 61. A class contains 13 girls and 11 boys. The teacher randomly selects four students. Determine the probability that all four students selected are girls. (ii) (iii) Find the probability that the student is a Year 2 Art student. Are the events A and B independent? Justify your answer. (6) Given that a History student is selected at random, calculate the probability that the student is in Year 1. Answers: Two students are selected at random from the college. Calculate the probability that one student is in Year 1, and the other in Year 2. (Total 12 marks) Residents of a small town have savings which are normally distributed with a mean of $3000 and a standard deviation of $500. (i) What percentage of townspeople have savings greater than $3200? (ii) Two townspeople are chosen at random. What is the probability that both of them have savings between $2300 and $3300? (iii) The percentage of townspeople with savings less than d dollars is 74.22%. Find the value of d. (Total 8 marks) 69 70

36 62. The following table shows the mathematics marks scored by students. Mark Frequency k The events A and B are independent such that P(B) = 3P(A) and P(A B) = Find P(B) The mean mark is 4.6. Find the value of k. Write down the mode. Answers: In the research department of a university, 300 mice were timed as they each ran through a maze. The results are shown in the cumulative frequency diagram opposite. Answers: How many mice complete the maze in less than 10 seconds? (1)

37 Estimate the median time. (1) Another way of showing the results is the frequency table below. Time t (seconds) Number of mice t < t < t < t < 10 p 10 t < 11 q 11 t < t < t < t < (i) Find the value of p and the value of q. (ii) Calculate an estimate of the mean time. 65. The following probabilities were found for two events R and S. P(R) = 3 1, P(S R) = 5 4, P(S R ) = 4 1. Copy and complete the tree diagram

38 Find the following probabilities. 67. Let A and B be independent events such that P(A) = 0.3 and P(B) = 0.8. (i) P(R S). Find P(A B). (ii) P(S). Find P(A B). (iii) P(R S). (7) (Total 10 marks) Are A and B mutually exclusive? Justify your answer. 66. The heights, H, of the people in a certain town are normally distributed with mean 170 cm and standard deviation 20 cm. A person is selected at random. Find the probability that his height is less than 185 cm. Given that P (H > d) = , find the value of d

39 68. The heights of a group of students are normally distributed with a mean of 160 cm and a standard deviation of 20 cm. A student is chosen at random. Find the probability that the student s height is greater than 180 cm. In this group of students, 11.9% have heights less than d cm. Find the value of d. 69. Consider the four numbers a, b, c, d with a b c d, where a, b, c, d. The mean of the four numbers is 4. The mode is 3. The median is 3. The range is 6. Find the value of a, of b, of c and of d

40 70. The population below is listed in ascending order. 5, 6, 7, 7, 9, 9, r, 10, s, 13, 13, t The median of the population is 9.5. The upper quartile Q 3 is 13. Write down the value of (i) r; (ii) s. The mean of the population is 10. Find the value of t. 71. The probability distribution of the discrete random variable X is given by the following table. x P(X = x) 0.4 p Find the value of p. Calculate the expected value of X

41 72. In a class, 40 students take chemistry only, 30 take physics only, 20 take both chemistry and physics, and 60 take neither. 73. The four populations A, B, C and D are the same size and have the same range. Frequency histograms for the four populations are given below. Find the probability that a student takes physics given that the student takes chemistry. Find the probability that a student takes physics given that the student does not take chemistry. State whether the events taking chemistry and taking physics are mutually exclusive, independent, or neither. Justify your answer. Each of the three box and whisker plots below corresponds to one of the four populations. Write the letter of the correct population under each plot Each of the three cumulative frequency diagrams below corresponds to one of the four populations. Write the letter of the correct population under each diagram

42 74. Three students, Kim, Ching Li and Jonathan each have a pack of cards, from which they select a card at random. Each card has a 0, 3, 4, or 9 printed on it. Kim states that the probability distribution for her pack of cards is as follows. x P(X = x) A game is played, where a die is tossed and a marble selected from a bag. Bag M contains 3 red marbles (R) and 2 green marbles (G). Bag N contains 2 red marbles and 8 green marbles. A fair six-sided die is tossed. If a 3 or 5 appears on the die, bag M is selected (M). If any other number appears, bag N is selected (N). A single marble is then drawn at random from the selected bag. Copy and complete the probability tree diagram on your answer sheet. Explain why Kim is incorrect. Ching Li correctly states that the probability distribution for her pack of cards is as follows. x P(X = x) 0.4 k 2k 0.3 Find the value of k. Jonathan correctly states that the probability distribution for his pack of cards is given by x +1 P(X = x) =. One card is drawn at random from his pack. 20 (i) Calculate the probability that the number on the card drawn is 0. (ii) Calculate the probability that the number on the card drawn is greater than 0. (Total 8 marks) (i) Write down the probability that bag M is selected and a green marble drawn from it. (ii) Find the probability that a green marble is drawn from either bag. (iii) Given that the marble is green, calculate the probability that it came from Bag M. A player wins $2 for a red marble and $5 for a green marble. What are his expected winnings? (Total 14 marks) (7) 76. In a large school, the heights of all fourteen-year-old students are measured. The heights of the girls are normally distributed with mean 155 cm and standard deviation 10 cm. The heights of the boys are normally distributed with mean 160 cm and standard deviation 12 cm. Find the probability that a girl is taller than 170 cm

43 Given that 10% of the girls are shorter than x cm, find x. 77. The box and whisker diagram shown below represents the marks received by 32 students. Given that 90% of the boys have heights between q cm and r cm where q and r are symmetrical about 160 cm, and q < r, find the value of q and of r. In the group of fourteen-year-old students, 60% are girls and 40% are boys. The probability that a girl is taller than 170 cm was found in part. The probability that a boy is taller than 170 cm is Write down the value of the median mark. Write down the value of the upper quartile. A fourteen-year-old student is selected at random. Estimate the number of students who received a mark greater than 6. (d) Calculate the probability that the student is taller than 170 cm. (e) Given that the student is taller than 170 cm, what is the probability the student is a girl? (Total 17 marks) 85 86

44 78. Events E and F are independent, with P(E) = 3 2 and P(E F) = 3 1. Calculate P(F); P(E F). 79. A fair coin is tossed five times. Calculate the probability of obtaining exactly three heads; at least one head

45 cumulative frequency 80. The heights of certain flowers follow a normal distribution. It is known that 20% of these flowers have a height less than 3 cm and 10% have a height greater than 8 cm. Find the value of the mean µ and the standard deviation σ. 81. The following is the cumulative frequency curve for the time, t minutes, spent by 150 people in a store on a particular day time ( t) (i) How many people spent less than 5 minutes in the store? (ii) (iii) Find the number of people who spent between 5 and 7 minutes in the store. Find the median time spent in the store. (6) 89 90

46 Given that 40% of the people spent longer than k minutes, find the value of k. 83. Consider the events A and B, where P(A) = 5 2, P(B ) = 4 1 and P(A B) = 8 7. Write down P(B). (i) On your answer sheet, copy and complete the following frequency table. (ii) t (minutes) 0 t < 2 2 t < 4 4 t < 6 6 t < 8 8 t < t < 12 Frequency Hence, calculate an estimate for the mean time spent in the store. (5) (Total 14 marks) Find P(A B). Find P(A B). 82. Two fair four-sided dice, one red and one green, are thrown. For each die, the faces are labelled 1, 2, 3, 4. The score for each die is the number which lands face down. Write down (i) (ii) the sample space; the probability that two scores of 4 are obtained. Let X be the number of 4s that land face down. Copy and complete the following probability distribution for X. x P(X = x) Find E(X). (Total 10 marks) 91 92

47 Cumulative frequency 84. The cumulative frequency graph below shows the heights of 120 girls in a school Given that 60% of the girls are taller than a cm, find the value of a Height in centimetres Using the graph (i) (ii) write down the median; find the interquartile range

48 85. The heights of boys at a particular school follow a normal distribution with a standard deviation of 5 cm. The probability of a boy being shorter than 153 cm is Calculate the mean height of the boys. Find the probability of a boy being taller than 156 cm. 86. The eye colour of 97 students is recorded in the chart below. Brown Blue Green Male Female One student is selected at random. Write down the probability that the student is a male. Write down the probability that the student has green eyes, given that the student is a female. Find the probability that the student has green eyes or is male

49 87. The weights of a group of children are normally distributed with a mean of 22.5 kg and a standard deviation of 2.2 kg. Write down the probability that a child selected at random has a weight more than 25.8 kg. 88. A set of data is 18, 18, 19, 19, 20, 22, 22, 23, 27, 28, 28, 31, 34, 34, 36. The box and whisker plot for this data is shown below. Of the group 95% weigh less than k kilograms. Find the value of k. Write down the values of A, B, C, D and E. A =... B =... C=... D =... E =... Find the interquartile range. The diagram below shows a normal curve. On the diagram, shade the region that represents the following information: 87% of the children weigh less than 25 kg 97 98

50 89. A pair of fair dice is thrown. Copy and complete the tree diagram below, which shows the possible outcomes. 90. There are 50 boxes in a factory. Their weights, w kg, are divided into 5 classes, as shown in the following table. Class Weight (kg) Number of boxes A 9.5 w < B 18.5 w < C 27.5 w < D 36.5 w < E 45.5 w < Show that the estimated mean weight of the boxes is 32 kg. There are x boxes in the factory marked Fragile. They are all in class E. The estimated mean weight of all the other boxes in the factory is 30 kg. Calculate the value of x. Let E be the event that exactly one four occurs when the pair of dice is thrown. An additional y boxes, all with a weight in class D, are delivered to the factory. The total estimated mean weight of all of the boxes in the factory is less than 33 kg. Find the largest possible value of y. (5) (Total 12 marks) Calculate P(E). The pair of dice is now thrown five times. Calculate the probability that event E occurs exactly three times in the five throws. 91. Two restaurants, Center and New, sell fish rolls and salads. Let F be the event a customer chooses a fish roll. Let S be the event a customer chooses a salad. Let N be the event a customer chooses neither a fish roll nor a salad. In the Center restaurant P(F) = 0.31, P(S) = 0.62, P(N) = (d) Calculate the probability that event E occurs at least three times in the five throws. (Total 12 marks) Show that P(F S) = Given that a customer chooses a salad, find the probability the customer also chooses a fish roll. Are F and S independent events? Justify your answer

51 At New restaurant, P(N) = Twice as many customers choose a salad as choose a fish roll. Choosing a fish roll is independent of choosing a salad. (d) Find the probability that a fish roll is chosen. (7) (Total 16 marks) Hence, calculate an estimate of the mean age. 92. The histogram below represents the ages of 270 people in a village. Use the histogram to complete the table below. Age range Frequency Mid-interval value 0 age < age < age < age < age

52 93. The Venn diagram below shows information about 120 students in a school. Of these, 40 study Chinese (C), 35 study Japanese (J), and 30 study Spanish (S). 94. A bag contains four apples (A) and six bananas (B). A fruit is taken from the bag and eaten. Then a second fruit is taken and eaten. Complete the tree diagram below by writing probabilities in the spaces provided. A student is chosen at random from the group. Find the probability that the student studies exactly two of these languages; (1) studies only Japanese; does not study any of these languages. Find the probability that one of each type of fruit was eaten

53 95. A discrete random variable X has a probability distribution as shown in the table below. 96. The weights of chickens for sale in a shop are normally distributed with mean 2.5 kg and standard deviation 0.3 kg. x A chicken is chosen at random. P(X = x) 0.1 a 0.3 b (i) Find the probability that it weighs less than 2 kg. Find the value of a + b. Given that E(X) =1.5, find the value of a and of b. (ii) (iii) Find the probability that it weighs more than 2.8 kg. Copy the diagram below. Shade the areas that represent the probabilities from parts (i) and (ii). (iv) Hence show that the probability that it weighs between 2 kg and 2.8 kg is (to four significant figures). A customer buys 10 chickens. (i) (ii) Find the probability that all 10 chickens weigh between 2 kg and 2.8 kg. Find the probability that at least 7 of the chickens weigh between 2 kg and 2.8 kg. (6) (Total 13 marks) (7) 97. A four-sided die has three blue faces and one red face. The die is rolled. Let B be the event a blue face lands down, and R be the event a red face lands down. Write down (i) (ii) P (B); P (R)

54 If the blue face lands down, the die is not rolled again. If the red face lands down, the die is rolled once again. This is represented by the following tree diagram, where p, s, t are probabilities. 98. A box contains 100 cards. Each card has a number between one and six written on it. The following table shows the frequencies for each number. Number Frequency k Calculate the value of k. Find (i) the median; Find the value of p, of s and of t. Guiseppi plays a game where he rolls the die. If a blue face lands down, he scores 2 and is finished. If the red face lands down, he scores 1 and rolls one more time. Let X be the total score obtained. 3 (i) Show that P (X = 3) =. 16 (ii) Find P (X = 2). (d) (i) Construct a probability distribution table for X. (ii) Calculate the expected value of X. (5) (ii) the interquartile range. (e) If the total score is 3, Guiseppi wins $10. If the total score is 2, Guiseppi gets nothing. (5) (Total 7 marks) Guiseppi plays the game twice. Find the probability that he wins exactly $10. (Total 16 marks)

55 99. There are 20 students in a classroom. Each student plays only one sport. The table below gives their sport and gender In a school with 125 girls, each student is tested to see how many sit-up exercises (sit-ups) she can do in one minute. The results are given in the table below. Football Tennis Hockey Female Male One student is selected at random. Number of sit-ups Number of students Cumulative number of students p (i) (ii) Calculate the probability that the student is a male or is a tennis player. Given that the student selected is female, calculate the probability that the student does not play football. Two students are selected at random. Calculate the probability that neither student plays football. 18 q (i) Write down the value of p. (ii) Find the value of q. Find the median number of sit-ups. Find the mean number of sit-ups. (Total 7 marks) (Total 7 marks)

56 101. A factory makes switches. The probability that a switch is defective is The factory tests a random sample of 100 switches The following table shows the probability distribution of a discrete random variable X. Find the mean number of defective switches in the sample. x Find the probability that there are exactly six defective switches in the sample. Find the probability that there is at least one defective switch in the sample. (Total 7 marks) P (X = x) k k Find the value of k. Find the expected value of X A box contains a large number of biscuits. The weights of biscuits are normally distributed with mean 7 g and standard deviation 0.5 g. One biscuit is chosen at random from the box. Find the probability that this biscuit (i) weighs less than 8 g; (ii) weighs between 6 g and 8 g. Five percent of the biscuits in the box weigh less than d grams. (i) Copy and complete the following normal distribution diagram, to represent this information, by indicating d, and shading the appropriate region. (ii) Find the value of d. (5) (Total 7 marks) The weights of biscuits in another box are normally distributed with mean µ and standard deviation 0.5 g. It is known that 20% of the biscuits in this second box weigh less than 5 g. Find the value of µ. (Total 13 marks)

57 104. The heights of certain plants are normally distributed. The plants are classified into three categories. The shortest 12.92% are in category A. The tallest 10.38% are in category C. All the other plants are in category B with heights between r cm and t cm. Complete the following diagram to represent this information Paula goes to work three days a week. On any day, the probability that she goes on a red bus is 1. 4 Write down the expected number of times that Paula goes to work on a red bus in one week. In one week, find the probability that she goes to work on a red bus on exactly two days; on at least one day. Given that the mean height is 6.84 cm and the standard deviation 0.25 cm, find the value of r and of t. (5) (Total 7 marks) (Total 7 marks)

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