Question Bank | Unit 9

1. $A = {2, 3, 6, 9, 11}$ and $B = {13, 14, 15, 17, 19, 21}$ .

(a) If one element is selected at random from $A$ , write down the probability that it is odd.

(b) If one element is selected from each set, calculate the probability that both elements are even, expressing your answer as a fraction in its lowest terms.

2. (a) There are 30 blue balls and $x$ red balls in a bag. A ball is drawn at random from the bag.

(i) Write down, in terms of $x$ , an expression for the probability that the ball drawn is red.

(ii) Given that this probability is $\frac{7}{13}$ , find $x$ .

(b) $M$ and $N$ are the midpoints of adjacent sides of the square $W X Y Z$ . A point is selected at random in the square. Calculate the probability that it lies in triangle $M Y N$ .

3. From a group of five children, consisting of three girls and two boys, one child is chosen at random. Write down the probability that the child chosen is a girl. A second child is then chosen from the remaining four children. Given that the first child chosen is a girl, write down the probability that the second child chosen is also a girl. On another occasion, two children are chosen at random from this same group of three girls and two boys. Calculate the probability that

(a) both are girls,

(b) both are boys,

(c) they are of different sexes.

4. A ball is dropped at random into one of the eight holes, numbered as shown in the diagram. The number under each hole gives the score obtained when the ball drops into the hole.

(a) State the probability of scoring 1.

(b) If the ball is dropped twice, find the probability of scoring

(i) a total of 6,

(ii) a total of 4.

5. The faces of a cubical die are numbered from one to six. Three such dice are used in an experiment. The first one is thrown and the number shown on the top face is a five. The second and third dice are then thrown. Calculate, as a fraction, the probability that

(a) all three numbers shown are odd,

(b) the sum of the three numbers is less than 10,

(c) the product of the three numbers is a prime number.

6. Two six-sided dice, one coloured black and one red, are thrown. Giving each answer as a fraction, calculate the probability that

(a) the score on the red die is 3,

(b) each die shows a score of 5,

(c) the score on the black die is either 2 or 4,

(d) either the score on the black die is 1, or the score on the red die is 6, but not both.

7. A bag contains four counters, one marked with the letter A, one with the letter B and two with L. The counters are drawn at random from the bag, one at a time, without replacement. In each of the following cases calculate the probability that

(a) the first two counters to be drawn out will each have the letter L marked on them,

(b) the second counter to be drawn out will be that with the letter B marked on it,

(c) the order in which the counters are drawn will spell out the word B A L L.

8. Each of the following experiments starts with four cards which are numbered 1, 4, 6 and 9 respectively.

(a) One card is chosen at random. State the probability that it is the card numbered 9.

(b) Two cards are chosen at random. Find the probability that the number on each card is even.

(c) Two cards are chosen at random. Find the probability that the numbers on these two cards add up to more than 9.

9. A boy has 40 records in a rack, 24 of which are "singles" and the remainder "long players". In the Venn diagram, $S$ and $L$ are the sets of "singles" and "long players" and the set $P$ , indicated by the dotted outline, is the set of "pop records". The numbers and letters represent the number of records in each subset. (For example, there are 20 singles which are "pop records".)

(a) If a "single" is chosen at random from the rack, find the probability that it is not a "pop record".

(b) If, instead, two records are taken at random from the rack, find the probability that they are both "pop singles".

(c) On another occasion, a record is chosen at random from the rack. The probability that this record is either a "long playing pop record" or any "single" is $\frac{17}{20}$ . Find the value of $m$ and the value of $n$ .

10. (a) There are 25 red marbles and $x$ blue marbles in a box. One marble is selected at random. Given that the probability that it is blue is $\frac{1}{6}$ , calculate the value of $x$ .

(b) In another box there are 27 green marbles and 13 yellow marbles. Two marbles are selected at random. Calculate the probability that both are green, giving your answer as a fraction in its lowest terms.

11. (a) In the diagram, the centre of the larger circle is $O$ , its radius is 9 cm and $P \hat{O} Q = 70^{\circ}$ . The radius of the smaller circle is 2 cm. A point is selected at random inside the larger circle. Expressing each answer as a fraction in its lowest terms, calculate the probability that the point lies (i) inside the sector $P O Q$ , (ii) inside the smaller circle.

(b) In a raffle, 100 tickets are sold of which 50 are red, 30 are blue and the rest are green. After the tickets are thoroughly mixed, one is drawn for the first prize and another is drawn for the second prize. Expressing each answer as a fraction in its lowest terms, find the probability that (i) the first ticket drawn is green, (ii) the first ticket drawn is red and the second ticket drawn is blue.

12. Two bags each contain six counters. In each bag there are three red counters, two white counters and one blue counter.

(a) One counter is drawn from the first bag. Find the probability that it is white.

(b) One counter is drawn from the second bag. Find the probability that it is not red. These counters are now returned to their original bags.

(c) One counter is drawn from each bag. Find the probability that the two counters are (i) both blue, (ii) each of a different colour.

13. A bag contains twelve sweets, eight of which are red and the other four are yellow. Angela takes a sweet at random and eats it. Hence the probability that she chooses a red sweet is $\frac{8}{12}$ as shown on the probability tree diagram below. Bruce then takes a sweet at random and eats it.

(a) Complete the probability tree diagram in the answer space.

(b) Find the probability that (i) both take yellow sweets, (ii) they take sweets of different colours.

14. Two six sided dice, one coloured red and the other blue, are thrown together. Giving each answer as a fraction, find the probability that

(a) the total score is 12,

(b) the total score is 5,

(c) the score on the red die is twice the score on the blue die.

15. On any visit to the grocers, the probability that Elizabeth will buy a tin of beans is $\frac{1}{6}$ and the probability that she will buy a packet of tea is $\frac{2}{3}$ .

(a) Find the probability that, on one particular visit, Elizabeth will not buy a tin of beans.

(b) Find the probability that, on three successive visits, she will buy a packet of tea each time.

16. Two bags contain coloured balls. Bag A contains 5 red balls, 2 white balls and 1 blue ball. Bag B contains 2 red balls and 3 blue balls.

(a) One ball is chosen at random from Bag A. Find the probability that it is not red.

(b) Two balls are chosen at random from Bag B. Find the probability that they are both blue.

(c) All the balls are replaced in their original bags and then one ball is chosen from each bag. Find the probability that one is red and one is white.

17. A survey is carried out on a group of 300 elderly people, 100 of whom are men. Find the probability that

(a) one person, chosen at random, will be a woman,

(b) any two people, chosen at random from the original group, will (i) both be men, (ii) not both be men.

18. (a) I throw an ordinary six sided die. Write down, as a fraction, the probability that the number shown on the die is even.

(b) There are 56 marbles in a bag. Given that the probability of choosing a red marble is exactly $\frac{5}{8}$ , calculate the number of marbles in the bag which are not red.

19. A drawer contains 6 grey socks and 9 blue socks. One dark morning two socks are taken at random one after the other from the drawer. By drawing a tree diagram, or otherwise, calculate the probability that

(a) the first sock is grey,

(b) both socks are grey,

(c) both socks are of the same colour.

20. (a) I throw an ordinary six sided die. Write down, as a fraction, the probability that the number shown on the die is odd.

(b) There are 63 marbles in a bag. Given that the probability of choosing a red marble at random is exactly $\frac{5}{9}$ , calculate the number of marbles in the bag which are not red.

21. A girl has 7 coins in her purse. She has six 10c coins and one 50c coin. She takes two coins at random from her purse, one after the other,

(a) Complete the probability tree diagram shown in the answer space.

(b) Find the probability that the total value of the two coins is

(i) 20c,

(ii) $1,

(iii) 60c.

22. $P = {- 3, - 2, - 1, 0}$ and $Q = {- 1, 0, 1, 2, 3}$ . An element is chosen at random from the set $P$ and is denoted by $x$ . An element is chosen at random from the set $Q$ and is denoted by $y$ . Find the probability of each of the following

(a) $x < 0$ ,

(b) $y^{2} = 1$ ,

(c) $x y \geq 0$ .

23. Two six sided unbiased dice are thrown together. Giving each of your answers as a fraction in its lowest terms, find the probability that

(a) the sum of the two numbers is 10,

(b) the two numbers are not equal,

(c) when the two numbers are multiplied together, the result is an even number.

24. Jane has six T-shirts; one is red, one blue, one yellow and three are white. She has three skirts; one is grey, one red and one blue. She selects a T-shirt and a skirt at random. Expressing each answer as a fraction in its lowest terms, find the probability that Jane selects

(a) a T-shirt which is not red,

(b) a white T-shirt and a grey skirt,

(c) a T-shirt and a skirt of the same colour.

25. On any day, the probability that I will oversleep is $\frac{1}{5}$ . Find the probability that

(a) I will not oversleep on a particular day,

(b) I will oversleep on two particular consecutive days,

(c) I will oversleep on just one of two particular consecutive days.

26. On any day, the probability that it will rain is $\frac{1}{6}$ . Find the probability that

(a) it will not rain on a particular day,

(b) it will rain on two particular consecutive days,

(c) it will rain on just one of two particular consecutive days.

27. The probability that Janice does not score a goal in any particular game of hockey is $\frac{2}{3}$ .

(a) Find the probability that in the first two games of the season (i) she does not score in either game, (ii) she scores in both games.

(b) The probability that Janice scores more than one goal in any particular game is $\frac{1}{12}$ . Find the probability that she scores exactly one goal in the last match of the season.

28. John has 5 red discs and 4 white discs in a bag. He takes two discs at random from the bag one after the other.

(a) Complete the tree diagram in the answer space.

(b) Find the probability that

(i) the first disc is red and the second is white,

(ii) the two discs have different colours,

(iii) the second disc chosen is white.

29. (a) My dog Ben is given 11 biscuits for his breakfast. 7 of them are black, 3 are red and 1 is yellow. (i) He eats one of them. Assuming that he is equally fond of each sort of biscuit, what is the probability that the biscuit he eats is red? (ii) He then eats a second biscuit. What is the probability that the first biscuit is red and the second is black?

(b) On another day he is again given 7 black biscuits, 3 red biscuits and 1 yellow biscuit. He eats only 2 of them. What is the probability that 1 is yellow and 1 black?

30. There are 22 balls in a snooker set. 15 of them are red, 1 is white and 6 are other colours. All 22 balls were placed in a bag and 1 of them was drawn out at random. The ball was not replaced. A second ball was then drawn out, also at random, and was not replaced. Some of the probabilities are shown in the tree diagram below.

(a) Calculate the values of $p$ , $q$ , $r$ and $s$ , shown on the tree diagram.

(b) Expressing each of your answers as a fraction, in its lowest terms, calculate the probability that (i) when two balls are drawn out, they will both be red, (ii) when two balls are drawn out, one will be red and one will be white.

(c) If a third ball is drawn out, calculate the probability that all 3 balls will be red.

(d) Each of the snooker balls is a sphere of diameter 5·25 cm and the density of the material used is 1·85 g/cm³. Calculate the total mass of a set of 22 snooker balls, giving your answer in kilograms, correct to 1 decimal place.
[Take $π$ to be 3·142. The volume of a sphere of radius $r$ is $\frac{4}{3} π r^{3}$ .]

31. Charles is sticking stamps to the value of 30 cents on each of a large number of envelopes. He has many 5 cent and 10 cent stamps with which to do this.

(a) Draw a small circle around each point in the diagram in the answer space which represents a possible choice of stamps he could make. (One small circle, representing the choice of one 10 cent stamp and four 5 cent stamps, has already been drawn for you.)

(b) When he has finished, Charles notices that there is an equal number of envelopes bearing each possible choice of stamps. He chooses one envelope at random. Find the probability that it has (i) exactly two 5 cent stamps, (ii) more 5 cent stamps than 10 cent stamps.

32. A bag contains 1 blue ball and 4 red balls. A girl takes two balls at random from the bag one after the other.

(a) Complete the probability tree diagram shown in the answer space.

(b) Find the probability that she has taken out

(i) two red balls,

(ii) one ball of each colour.

33. There are 3 Science books and 1 English book on a table. Joan selects two of the books at random, one after the other.

(a) Complete the probability tree diagram shown in the answer space.

(b) Find the probability that she has selected

(i) two Science books,

(ii) one book of each subject.

34. An Australian has three 50 cent coins and two 10 cent coins in his pocket. He takes coins out of his pocket, at random, one after the other. The coins are not replaced. The tree diagram below shows the possible outcomes and their probabilities.

(a) Find the value of $x$ .

(b) Find the probability that the total value of the two coins taken out is (i) 20 cents, (ii) 60 cents.

(c) The Australian takes out a third coin. Find, showing your working clearly, the probability that the total value of the three coins taken out is 70 cents.

35. The probability that a small child will eat all her breakfast on any morning is $\frac{4}{5}$ .

(a) (i) What is the probability that she will eat all her breakfast on both Wednesday and Thursday next week?

(ii) What is the probability that she will eat all her breakfast on either Wednesday or Thursday next week, but not on both days?

(b) Write down, but do not evaluate, an expression for the probability that she will eat all her breakfast on every day next week.

36. When a girl wakes up each morning, the probability that she feels happy is $\frac{9}{10}$ .

(a) (i) What is the probability that she will feel happy when she wakes up, on both Monday and Tuesday next week?

(ii) What is the probability that she will feel happy when she wakes up, on either Monday or Tuesday next week, but not on both days?

(b) Write down, but do not evaluate, an expression for the probability that she will wake up happy on every day of next week.

37. The probability that a football team will win any particular game is $\frac{3}{4}$ .

(a) Find the probability that the team (i) will not win the first game of the season, (ii) will not win either of the first two games of the season.

(b) The probability that the team will draw any particular game is $\frac{1}{6}$ . Find the probability that the team will lose the last game of the season.

38. A bag contains 6 red sweets, 3 yellow sweets and 1 green sweet. Two sweets are drawn at random from the bag, one after the other, and are not replaced. Expressing your answer as a fraction, find the probability that

(a) the first sweet taken is red,

(b) both of the sweets are red,

(c) both of the sweets are the same colour,

(d) the two sweets are of different colours.

39. Centup is a dice game for two or more players. To play Centup an ordinary die numbered 1, 2, 3, 4, 5 and 6 is used. The rules are

1 When it is his turn, a player may throw the die as many times as he likes, or until a '1' is thrown.

2 If a '1' is thrown the player's total is zero for that turn.

3 If he decides to stop before throwing a '1', the total of the numbers thrown during his turn is added to his score.

4 The first player whose score reaches (or passes) 100 is the winner.

For example, David's score is 49. He throws '2', '6' and decides to stop. His total for this turn is 8 and his score becomes 57.

John's score is 55. He throws '3', '5', '4', '1', so that his score remains at 55.

Suppose that your score at Centup is now 96 and it is your turn.

(a) What is the probability that you will win with just one throw of the dice?

(b) Suppose that you take two throws to win. One way of doing this is to throw first a '2', and then a '3'. This can be written [2, 3]. List all the possible outcomes, including [2, 3], which lead to a win in exactly two throws?

(c) What is the probability that you will win with exactly two throws?

(d) Suppose that you decide to keep throwing until either you have thrown a 'I' or have won. What is the probability that you will not win the game during this turn?

40. A teacher (T), 3 boys (B) and 4 girls (G) are on a school committee. Two representatives are selected at random from the committee.

(a) Complete the tree diagram on the right.

(b) Calculate the probability that

(i) both representatives are girls,

(ii) the representatives are not both girls,

(iii) one representative is a boy and the other a girl.

41. A bag contains a number of balls each one coloured either red or blue. A ball is chosen at random and then put back into the bag. This process is repeated several times.

(a) The probability of choosing a red ball is $r$ . Write down, in terms of $r$ , the probability of choosing a blue ball.

(b) The tree diagram below represents the situation when the process has been carried out twice. Expressing your answer in terms of $r$ , find the probability that a red ball was chosen each time.

(c) The process was carried out nine times. Find the probability that

(i) a red ball was chosen every time,

(ii) at least one blue ball was chosen.

42. A number of balls, coloured either black or white, are contained in a bag. A ball is chosen at random and then put back into the bag. This process is repeated several times.

(a) The probability of choosing a black ball is $p$ . Write down, in terms of $p$ , the probability of choosing a white ball.

(b) The tree diagram represents the situation when the process has been carried out twice. Expressing your answer in terms of $p$ , find the probability that a black ball was chosen each time.

(c) The process was carried out ten times. Find the probability that

(i) a black ball was chosen every time,

(ii) at least one white ball was chosen.

43. A box of chocolates contained 6 chocolates with hard centres and 4 chocolates with soft centres. Ann took a chocolate, selected at random, from the box and ate it. Bruce then took a chocolate, selected at random, from the box. Expressing each answer as a fraction in its simplest form, find the probability that

(a) Ann took a chocolate with a hard centre,

(b) Ann took a chocolate with a hard centre and Bruce took a chocolate with a soft centre,

(c) both Ann and Bruce took a chocolate with a soft centre,

(d) Bruce took a chocolate with a soft centre.

44. An unbiased Blue die is numbered 1, 9, 10, 11, 12 and 13.

An unbiased Red die is numbered 2, 3, 4, 14, 15 and 16.

The possibility diagram when the two dice are thrown is shown below.

The letter $R$ indicates that the number (14) thrown on the Red die is greater than the number (9) thrown on the Blue die.

(a) Copy the diagram. Put a letter $R$ in each square where the number thrown on the Red die is greater than the number thrown on the Blue die.

(b) Show that, when the two dice are each thrown once, the probability that the number thrown on the Red die is greater than the number thrown on the Blue die is $\frac{7}{12}$ .

(c) The two dice are each thrown twice. By using a tree diagram, or otherwise, calculate the probability that the number thrown on the Red die is greater than the number thrown on the Blue die

(i) both times,

(ii) just once.

(d) An unbiased Green die is numbered 5, 6, 7, 8, 17 and 18.

Showing your working clearly, find out whether the Red or the Blue die is more likely to show a number greater than the number thrown on the Green die.

45. In a game two dice are used. Die A has 2 red faces and 4 white faces. Die B has 4 red faces and 2 blue faces. The two dice are thrown together.

(a) By using the tree diagram below, or otherwise, find the probability that

(i) both dice show a red face on top,

(ii) just one die shows a red face on top.

(b) If both dice show red, the player wins a prize. If just one die shows red, the player throws both dice again. He wins a prize if both show red this time. Calculate the probability that the player wins a prize on either the first throw or the second throw.

46. (a) In this part of the question all probabilities should be given as exact decimals.

The ticket machine in a car park takes 50 cent coins and $1 coins. A ticket costs $1·50. The probability that the machine will accept a particular 50 cent coin is 0·9 and that it will accept a particular $1 coin is 0·8.

(i) What is the probability that the machine will not accept a particular 50 cent coin?

(ii) Leslie put one 50 cent coin and one $1 coin into the machine. Calculate the probability that the machine will not accept either of these coins.

(iii) Joan only has three 50 cent coins. Calculate the probability that

(a) the machine will accept all three coins,

(b) Joan will not get a ticket.

(b) The probability that Robin goes to work by car on any particular day is $\frac{3}{4}$ . The probability that Samantha goes to work by car on any particular day is $\frac{2}{3}$ . Calculate, as a fraction, the probability that next Tuesday just one of them will go to work by car.

47.

The diagram shows a grid of squares. A button is placed on one of the squares. A fair die is thrown. If 1, 2, 3 or 4 is thrown, the button is moved one square to the left. If 5 or 6 is thrown, the button is moved one square to the right.

(i) The button is placed on square $X$ . The die is thrown once. What is the probability that the button is moved to the right?

(ii) On another occasion the button is placed on square $Y$ . The die is thrown once and the button is moved. The die is thrown a second time and the button is moved again. Find the probability that the button finishes

(a) at $P$ ,

(b) at $Q$ ,

(c) at $Y$ ,

(d) at $P$ or $Q$ or $Y$ .

48. A packet contains a large number of flower seeds which look identical, but produce flowers with one of three colours, white, yellow or red.

One half of the seeds produce white flowers and one third produce yellow flowers.

The remainder of the seeds produce red flowers.

(a) Explain why the probability that a particular seed will produce a red flower is $\frac{1}{6}$ .

(b) Find the probability that a particular seed will produce a flower that is not yellow.

(c) Two seeds are planted

(i) Draw a tree diagram to show the possible outcomes and their probabilities.

(ii) Find the probability that

(a) both will produce a yellow flower,

(b) both will produce a blue flower,

(c) one will produce a yellow flower and the other a white flower,

(d) neither will produce a red flower.

49. When a particular die is thrown, the probability of a score of six is $\frac{1}{3}$ . The probabilities of scores of two, three, four and five are each $\frac{1}{6}$ .

(a) Find the probability of scoring one.

(b) Explain the significance of the answer to part (a).

(c) The die is thrown twice and the sum of the two scores is 6.

(i) Write down the possible scores on the two throws.

(ii) Calculate the probability that the sum of the two scores is 6.

50. There were 12 girls and 3 boys in a group of children. One child was chosen at random from the group. Another child was chosen at random from the remaining children. Expressing each answer as a fraction in its simplest form, calculate the probability that

(a) the first child chosen was a girl,

(b) the first child chosen was a girl and the second was a boy,

(c) a child of each sex was chosen.

51. A bag contains 15 identical discs. There are 8 red, 4 blue and 3 white discs. A disc is picked out at random and not replaced. A second disc is then picked out at random and not replaced. The tree diagram below shows the possible outcomes and some of their probabilities.

(a) Calculate the values of $p$ , $q$ , $r$ and $s$ shown on the tree diagram.

(b) Expressing each of your answers as a fraction in its lowest terms, calculate the probability that

(i) both discs will be red,

(ii) one disc will be red and the other blue.

(c) A third disc is now picked out at random. Calculate the probability that none of the three discs is white.

52. The probability that Catherine oversleeps is 0·4.

If she oversleeps, the probability that she cycles to school is 0·7.

If she does not oversleep, the probability that she cycles to school is 0·1.

(a) Complete the tree diagram, in the answer space, to represent this information.

(b) Calculate the probability that Catherine cycles to school.

53. A bag contains 3 black and 2 white balls. Two balls are taken from the bag at random, without replacement. By drawing a tree diagram, or otherwise, calculate the probability that

(a) both balls are black,

(b) at least one ball is white,

(c) the two balls are the same colour.

54. A bag contains 3 discs numbered 1 and 2 discs numbered 7. Two discs are taken from the bag at random without replacement. By drawing a tree diagram, or otherwise, calculate the probability that

(a) both discs are numbered 1,

(b) at least one disc is numbered 7,

(c) the sum of the two numbers on the discs is 8.

55. An examination is set every month. John takes the examination each month until he passes. Each time he takes the examination, the probability that he passes is $0.9$ .

(a) Find the probability that John

(i) fails the first examination and passes the second,

(ii) passes the examination in either the first or second month,

(iii) fails the first three examinations,

(iv) passes the examination in one of the first four months.

(b) (i) Find the probability, in terms of $n$ , that John fails the first $n$ examinations.

(ii) Write down the probability that John passes the examination in one of the first $n$ months.

56. Answer the whole of this question on a sheet of graph paper.

The length of time taken by 80 drivers to complete a journey is given in the table below.

Time ( $t$ minutes)	$60 < t \leq 80$	$80 < t \leq 90$	$90 < t \leq 95$	$95 < t \leq 100$	$100 < t \leq 110$	$110 < t \leq 130$
Number of drivers	4	10	14	20	24	8

(a) Using a scale of 2 cm to represent 10 minutes, draw a horizontal axis for times between 60 minutes and 130 minutes.

Choose a suitable scale for the vertical axis and draw a histogram to represent the information in the table.

(b) In which interval does the median of the distribution lie?

(c) Calculate an estimate of the mean time taken to complete the journey.

(d) One driver is chosen at random.

Expressing your answer as a fraction in its lowest terms, calculate the probability that she took 90 minutes or less for the journey.

(e) Two drivers are chosen at random. Expressing each answer as a fraction in its lowest terms, calculate the probability that

(i) both took more than 110 minutes for the journey,

(ii) one took 80 minutes or less for the journey and the other took more than 110 minutes.

Unit 9

Probability