(A) the Letter Chosen Is M . (B) the Letter Chosen Is Not E

1. There are 38 students in S3A and Herbert is one of the students of the class. Mr. Chan chooses a student at random to be the class monitor, what is the probability of choosing Herbert?

2. There are 24 members in a choir, of which 3 of them are aged under 16. If the choir leader chooses a member at random, find the probability that the member chosen is aged under 16.

3. If a letter is chosen at random from the word ‘SUPPLEMENT’, find the probability of each of the following events happening. (a) The letter chosen is ‘M’. (b) The letter chosen is not ‘E’.

4. A card is drawn at random from a pack of 52 playing cards (without jokers). Find the probability of drawing a card which satisfies each of the following conditions. (a) A black card (b) A face card

[ A face card means K, Q or J. ]

5. The following cumulative frequency curve shows the results of 80 students in an English Language examination. It is known that students fail the examination if their scores are below 50. If a student is chosen at random, find the probability of choosing a student who passes the examination.

c 60 n e u q

e 50 r f

e v i t 40 a l u m

u 30 C

0 10 20 30 40 50 60 70 80 90 100 Score

6. A card is drawn at random from a pack of 52 playing cards (without jokers). Find the probability of drawing a card which satisfies each of the following conditions. (a) A red 10 (b) A diamond or an ace (c) A black card or a king

7. There are 56 students in a music centre. 24 students learn to play pianos only, 22 students learn to play violins only. The rest of them learn to play both pianos and violins. If a student is chosen at random, find the probabilities of the following events happening. (a) The student learns to play both piano and violin. (b) The student learns to play only one kind of musical instruments.

8. The following cumulative frequency polygon shows the results of a group of students in a Mathematics examination. It is known that students who have scored between 60 and 80 get a grade B each. If a student is chosen at random, find the probability that the student chosen gets a grade B.

c 70 n e u q

e 60 r f

e v i t 50 a l u m

u 40 C

0 10 20 30 40 50 60 70 80 90 100 Score 9. There are 3 red dishes, 5 yellow dishes and n green dishes in a box. If a dish is drawn at random from 4 the box, the probability of getting a red dish or a yellow dish is , find the value of n. 7

10. There are 36 students in S3D, and the number of boys is 4 more than that of girls. If a student is selected at random, what is the probability that the student is a girl?

11. There are some black balls and white balls in a bag, of which the number of white balls is 5 more than that of black balls. If a ball is drawn at random from the bag, the probability of getting a black ball 3 is , find the number of black balls. 11

12. There are some white hats and red hats in a wardrobe, of which the number of white hats is less than that of red hats by 4. If a hat is drawn at random from the wardrobe, the probability of getting a red 5 hat is . Find the total number of hats in the wardrobe. 9

13. In S3A, 24 students wear glasses. If a student is selected at random from the class, the probability of 8 selecting a student with glasses is , find the number of students in S3A. 13

 2010 Chung Tai Educational Press. All rights reserved. 14. In a box of toy trains, x of them are manufactured by machine A and the rest are manufactured by machine B. If a toy train is chosen at random from the box, the probability of choosing a toy train 5 manufactured by machine A is . Express the number of toy trains manufactured by machine B in 8 terms of x.

15. Among 100 lucky draw tickets provided at the purchase of soft drinks, the prizes of 10 of them are a 1 radio each. To let the probability of selecting a ticket at random with a prize of a radio be , how 7 many additional tickets with prizes of a radio each should be provided?

1 16. For a batch of raffle tickets, the probability of winning a prize is . If 240 more tickets are added 30 1 without prizes, the probability of winning a prize becomes . Find the original number of raffle 90 tickets. 17. A factory produces three kinds of canned juice daily, where 20% of the canned juice produced is grape juice (G), 45% is orange juice (O) and the rest is apple juice (A). (a) If a can of juice is selected at random, what is the probability of selecting a can of apple juice? (b) It is given that the factory produces 14 000 cans of apple juice daily. Find the total number of cans of juice produced by the factory daily. (c) Hence find the number of cans of orange juice produced by the factory daily.

18. In a factory, there are three light bulb production lines, A, B and C. If a bulb is chosen at random, the 5 2 probabilities of choosing a bulb produced by production lines A and B are and respectively. 12 5 (a) If production line A produced x bulbs, express the number of bulbs produced by production line B in terms of x. (b) If production line C produced 440 bulbs, find the number of bulbs produced by production line A.

If a citizen is selected at random from the region, estimate the probabilities of the following events happening. (a) The blood type of the citizen is O. (b) The blood type of the citizen is not AB.

20. A candy manufacturer recorded the number of marshmallows in 100 packets and the results are as follows: Number of marshmallows 40 41 42 43 44 45 46 in each packet Frequency 14 10 12 15 16 14 19

(a) Find the experimental probability of getting a packet with 42 marshmallows. (b) If any packet with less than 42 marshmallows is below standard, find the experimental probability of getting a standard packet marshmallows. 21. The following table shows the distribution of the intelligence quotient (I.Q.) of S3 students of a school in the past three years. I.Q. Number of students 86 - 90 5 91 - 95 70 96 - 100 206 101 - 105 214 106 - 110 108 111 - 115 2

If a student is chosen at random from the school this year, estimate the probability of each of the following events happening. (a) The student’s I.Q. is between 95.5 and 105.5. (b) The student’s I.Q. is 105.5 or above.

 2010 Chung Tai Educational Press. All rights reserved. 22. The following cumulative frequency table shows the number of sleeping hours of Derek each night during a month. Sleeping hours less than 5 6 7 8 9 10 11 Cumulative Frequency 1 4 10 22 29 29 30

(a) Estimate the probability that the number of sleeping hours of Derek tonight is less than 8 hours. (b) Estimate the probability that the number of sleeping hours of Derek tonight is between 7 hours and 9 hours.

23. The following cumulative frequency polygon shows the distribution of the weight of the students in S3C. Weights of S3C students y

c 40 n e u q

e 30 r f

e v i t 20 a l u m

u 10 C

0 30 40 50 60 70 80 Weight (kg) If a student is chosen at random from S3, estimate the probability of each of the following events happening.

(a) The weight of the student is between 40 kg and 60 kg. (b) The weight of the student is 50 kg or above. 24. The table below shows the distribution of the members of a track and field team of a school last year. Boys Girls

S1 15 9 S2 11 7 S3 10 8 S4 6 2 S5 4 0

A member is selected at random from the team this year. Estimate the probability of each of the following events happening. (a) The member is a S1 boy. (b) The member is a girl. (c) The member is a S4 or S5 student.

 2010 Chung Tai Educational Press. All rights reserved. 25. The following frequency distribution table shows the weights of 100 moon cakes measured by a moon cake manufacturer in a survey. Weight (g) 201 - 210 211 - 220 221 - 230 231 - 240 241 - 250 251 - 260 Frequency 11 13 21 16 18 21

(a) Find the experimental probability of each of the following events happening. (i) The weight of a moon cake falls into the class interval 221 g - 230 g. (ii) The weight of a moon cake is between 210.5 g and 240.5 g. (iii) The weight of a moon cake is 230.5 g or above.

(b) If there are 20 000 moon cakes, estimate the number of moon cakes whose weights are 230.5 g or above.

26. From 1 600 eggs, x of them are chosen at random, of which 4 are rotten. 1 (a) If the relative frequency of the rotten eggs to the chosen ones is , find the value of x. 8 (b) According to the situation above, estimate the number of rotten eggs among these 1 600 eggs.

27. From 1 200 copies of a new book, 50 of them are chosen at random, of which x copies are misprinted. 1 (a) If the relative frequency of the misprinted copies to the chosen ones is , find the value of x. 25 (b) According to the situation above, estimate the number of copies without misprints among these 1 200 copies. 28. There are a total of 40 red, green and black ballpoint pens in a box. A ballpoint pen is drawn at random from the box with its colour recorded and then put back into the box. This process is repeated 1 000 times, and the records are as follows:

Colour of the ballpoint pen Red Green Black Frequency 247 125 628

(a) Find the experimental probability of each of the following events happening. (i) A black ballpoint pen is drawn. (ii) The ballpoint pen drawn is not green. (b) According to the conditions above, estimate the number of ballpoint pens of each colour in the box.

 2010 Chung Tai Educational Press. All rights reserved. 29. To investigate the number of fish in a pond, scientists caught 300 fish and put a ring around their tails before letting them go. After a period of time, the scientists caught another 300 fish again from the pond and discovered that 10 of them were with rings. (a) Find the relative frequency of the fish in the pond with rings. (b) Estimate the number of fish in the pond.

30. The following frequency distribution table shows the result of telephone interviews with 1 000 people about the paid channel they watch most frequently in the evening.

Channel Drama channel Entertainment News channel Movie channel Others channel Frequency 250 x 2x 200 100

(a) Find the value of x. (b) If a person is chosen at random, find the experimental probability of each of the following events happening. (i) The person watches the news channel most frequently. (ii) The person watches the drama channel or entertainment channel most frequently. (c) If you want to advertise on one of the paid channels in the evening, which channel will you choose? Explain your answer.

(d) Given that there are 300 000 people watching the paid channel in the evening, estimate the number of people who watch the news channel most frequently.

31. There are $20, $50 and $100 banknotes in a bag, and 20 of them are $50 banknotes. A banknote is drawn at random from the bag with its amount recorded and then put back into the bag. This process is repeated for a number of times, and the records are as follows: Amount of the banknote $20 $50 $100 Frequency 288 200 112

(a) Find the experimental probability of each of the following events happening. (i) $20 banknote is drawn. (ii) $50 banknote is drawn. (iii) $100 banknote is drawn. (b) Estimate the total number of banknotes in the bag. (c) Estimate the number of $20 banknotes in the bag.

 2010 Chung Tai Educational Press. All rights reserved. 32. There are 1 red ball (R), 1 white ball (W) and 1 green ball (G) in bag A, and 1 green ball (G), 1 purple ball (P) and 1 red ball (R) in bag B. If a ball is drawn randomly from each bag, find the probabilities of the following events happening. (a) One red ball and one purple ball are drawn. (b) Two white balls are drawn. (c) Two balls of the same colour are drawn.

33. It is given that the probabilities of giving birth to a baby boy (B) and a baby girl (G) are the same. Vivian has 3 children. (a) Use a tree diagram to list the sample space about the sex of these 3 children. (b) Find the total number of possible outcomes. (c) Find the probability that Vivian has 2 sons and 1 daughter.

34. Two fair dice are rolled. Find the probabilities of the following events happening. (a) Only one of the numbers obtained is 2. (b) At least one of the numbers obtained is a multiple of 3. 35. Two fair dice are rolled. Which of the following events is more likely to happen? Event A: The two numbers obtained are consecutive numbers. Event B: The two numbers obtained are the same.

36. A letter is chosen at random from each of the words ‘ABILITY’ and ‘DISABLE’.

(b) Find the probability of each of the following events happening. (i) The two letters are the same. (ii) The two letters are consonants. (iii) At least one of the letters is ‘I’.

37. Two fair dice are rolled. Find the probabilities of the following events happening. (a) The difference obtained by subtracting the smaller number from the larger one equals 3. (b) The product of the numbers obtained is less than 9.

38. The figure shows a road map from Town A to Town B. Macy and Kenneth both select a route at random to drive from Town B Town A to Town B. Find the probabilities of the following events happening. (a) Both of them select Highway C. (b) They select the same route. (c) None of them select Tunnel D. Town A

39. There are 2 red balls (R1 , R 2 ) , 2 black balls (B1 , B2 ) and 3 white balls (W1 , W2 , W3) in a bag. Two balls are drawn randomly. (a) List the sample space in the following table.

II R R B B W W W I 1 2 1 2 1 2 3

(b) Find the probability of each of the following events happening. (i) Two black balls are drawn. (ii) Two white balls are drawn. (iii) Two balls of the same colour are drawn. (iv) One black ball and one white ball are drawn.

40. There are 2 red scarves (R1 , R 2 ) and 3 white scarves (W1 , W2 , W3) in a drawer. Mary takes a scarf out at random and then puts it back. Then she takes a scarf out at random again. (a) Find the probability that the same scarf is taken out twice. (b) Find the probability that both scarves taken are in the same colour.

41. May has three $20 banknotes, two $50 banknotes and one $100 banknote inside her wallet. Two banknotes are taken out at random from the wallet. Find the probabilities of the following events happening.

(a) The face values of these two banknotes are the same. (b) The total amount of these two banknotes is $100. (c) The total amount of these two banknotes is more than $90. 42. There are a number of 3-digit numbers with digits of either ‘2’ or ‘3’. (a) List all 3-digit numbers that satisfy the above condition. (b) If a number is chosen at random from the numbers obtained in (a), find the probability of each of the following events happening. (i) Only one of the digits is ‘2’. (ii) The number is an even number.

 2010 Chung Tai Educational Press. All rights reserved. 43. There are a total of 6 purple and yellow balls in a bag, and the number of purple balls is twice the number of yellow balls. If two balls are drawn at random from the bag, find the probabilities of the following events happening. (a) Two purple balls are drawn. (b) Two balls in different colours are drawn.

44. There are 1 red ball and 2 white balls in a bag. After a ball is drawn at random, 1 red ball and 1 white ball are put into the bag, and then a ball is drawn at random again. (a) Find the probability of getting two balls in different colours. (b) Find the probability of getting a white ball first and then a red ball.

45. In each of the bags I and II, there are 1 white ball and 1 black ball. A ball is drawn at random from bag I and put into bag II, then a ball is drawn at random from bag II. (a) Find the total number of possible outcomes. (b) Find the probabilities of the following events happening. (i) The same ball is drawn twice. (ii) The balls drawn are of the same colour.

46. There are a number of 3-digit numbers formed by 1, 6 and 8, where the digits in the numbers are all different. (a) List the sample space. (b) If a number is chosen at random from the numbers obtained in (a), find the probabilities of the following events happening. (i) The number is greater than 650. (ii) The number is a multiple of 7. (iii) The number is a prime number.

 2010 Chung Tai Educational Press. All rights reserved. 47. The figure shows a card equally divided into four regions I, II, III and IV. If a point is marked on the card at random by a pin without touching the boundaries, find the probabilities of the following events happening.

IV III

(a) The point locates in region I. (b) The point does not locate in region I.

48. The figure shows a square tile with sides of 10 cm each. There is a circular region with the radius of 2 cm on the tile. A marble is rolling freely on the square tile until it stops. Given that the marble does not stop at the boundaries, find the probability that it does not stop at the circular region. (Express your answer in terms of .) 10 cm

2 cm 10 cm

49. Alan has three $20 banknotes, four $50 banknotes and two $100 banknotes in his wallet. If he picks out a banknote at random, find the expected value of the amount of the banknote.

50. In a football match, the winning team will score 3 points and the losing team will score nothing. If two teams draw, each of them will score 1 point. The probabilities for Tiger team to win, draw and lose in a match are 0.4, 0.3 and 0.3 respectively. (a) Find the expected value of the scores obtained by the team in each match. (b) Estimate the total scores obtained by the team after 20 matches.

51. As shown in the figure, Grace arrives at school between 8:15 a.m. and 9:00 a.m. to sit for an examination.

8:15 a.m. 8:30 a.m. 8:45 a.m. 9:00 a.m.

Time (a) If the examination starts at 8:30 a.m., find the probability that Grace is late for the examination. (b) If a candidate is late for the examination for 15 minutes or above, he or she cannot sit for the examination. Find the probability that Grace is late but can still sit for the examination.

52. A circular dartboard with the diameter of 20 cm is shown with a square inscribed in it. Suppose a dart hits a point randomly on the circular dartboard without hitting any boundaries, find the probability that it hits the square region. (Express your answer in terms of .)

53. A circular dartboard is shown in the figure. A rectangle of dimensions 40 cm  80 cm is inscribed in the dartboard. If a dart hits a point randomly on the circular dartboard without hitting any boundaries, find the probability that it hits the shaded region. (Express your answer in terms of .)

54. There are 192 students in S3 of a school. 36 of them are 14 years old, 108 of them are 15 years old and the rest are 16 years old. If a student in S3 is selected at random, find the expected value of the age of the student. 55. The following cumulative frequency polygon shows the results of S3B students in a Mathematics test.

c 40 n e u q

e 30 r f

e v i t 20 a l u m

u 10 C

0 20 40 60 80 100 Score

(a) Complete the following table. Score Class mark Frequency

20  x  40

40  x  60

60  x  80

80  x  100

(b) If a student is selected randomly from S3B, find the expected value of the score of the student in the Mathematics test.

56. In a game, a player requires to toss two fair coins in each round. If two heads are shown, the player will get 10 points. If two tails are shown, 20 points will be deducted. If one head and one tail are shown, the player will get 5 points. Find the expected value of the points obtained after ten rounds.

57. The number of members of each family in an estate and their corresponding probabilities are as follows: Number of family members 1 2 3 4 5 6 7 or above

Probability 0.04 x 0.28 0.35 0.14 0.01 0

(a) Find the value of x. (b) Find the expected value of the number of members of a family in the estate. 58. The figure shows a dartboard divided into three regions, A, B and C. O is the centre of the dartboard. Now, a dart hits a point at random on the dartboard without hitting the boundaries. If the dart hits

A 5x x O C B

(a) Find the value of x. (b) Find the expected value of the points scored for each dart thrown.

59. The figure shows a circular dartboard formed by two concentric circles with radii of 8 cm and 16 cm. Now, a dart hits a point at random on the dartboard without hitting the boundaries. If the dart hits region I or II, 25 points will be scored. If the dart hits region III or IV, 5 points will be scored. No points will be scored if the dart hits the white regions.

(a) Find the respective probabilities of hitting regions I, II, III and IV. (b) Find the expected value of the points scored for each dart thrown.

60. There are 15 multiple choice questions in an examination paper. Each question has 4 options in which only 1 of them is correct. x marks will be given for each correct answer, and y marks will be deducted for an incorrect one. If a student answers all the questions by choosing the options randomly, the expected value of his score will be 0. (a) Find x : y. (b) If the full mark of the examination paper is 90, find the values of x and y.

 2010 Chung Tai Educational Press. All rights reserved. 61. During a period of time, the waiting time of the people at a bus stop and their corresponding probabilities are as follows: Waiting time (minutes) 0 1 2 3 4 5 6 or above

Probability 0.05 0.15 0.35 x 0.1 y 0

If the expected value of the waiting time is 2.4 minutes, find the values of x and y.

62. The figure shows a circular dartboard formed by three concentric circles with radii of x cm, 12 cm

and z cm, where x  12 and z  12 . If a dart hits region I, 1 point will be scored. If a dart hits region III, 4 points will be deducted. No points will be scored if a dart hits region II. If a dart hits a point at random on the dartboard without hitting the boundaries, the expected value of the points scored is 0.

(a) Express x in terms of z. (b) Find all the possible integral values of x and z.

63. The following is a game at an amusement park.

4 identical 5 cm  5 cm squares are drawn on a 20 cm  20 cm table. A participant needs to throw a token with the diameter of 2 cm onto the table. If the token lies on a square without touching the boundaries, 10 tokens are given. If the centre of the token thrown does not fall onto the table, the token will drop to the ground and the participant can play once again.

(a) If a token lies on square ABCD without touching the boundaries of square ABCD, what is the area of the region formed by the possible locations of the centre of the token? (b) Find the expected value of the number of tokens given in each game. (c) Will you play the game? Explain briefly.