1. The table below shows the relative frequencies of the ages of the students at Ingham High School.

Age(in years)

Relativefrequency

13 0.11

14 0.30

15 0.23

16 0.21

17 0.15

Total 1

(a) If a student is randomly selected from this school, find the probability that

(i) the student is 15 years old;

(ii) the student is 16 years of age or older.

There are 1200 students at Ingham High School.

(b) Calculate the number of 15 year old students.

Working:

Answers:(a) (i) ……………………………………...

(ii) ……………………………………..(b) …………………………………………..

(Total 4 marks)

1

2. Note: For this question, it is important that you show your working and explain your method clearly.

A box contains 10 coloured light bulbs, 5 green, 3 red and 2 yellow. One light bulb is selected at random and put into the light fitting of room A.

(a) What is the probability that the light bulb selected is

(i) green?(1)

(ii) not green?(1)

A second light bulb is selected at random and put into the light fitting in room B.

(b) What is the probability that

(i) the second light bulb is green given the first light bulb was green?(l)

(ii) both light bulbs are not green?(2)

(iii) one room has a green light bulb and the other room does not have a green light bulb?

(3)

A third light bulb is selected at random and put in the light fitting of room C.

(c) What is the probability that

(i) all three rooms have green light bulbs?(2)

(ii) only one room has a green light bulb?(3)

(iii) at least one room has a green light bulb?(2)

(Total 15 marks)

2

3. A bag contains 2 red, 3 yellow and 5 green sweets.

Without looking, Mary takes one sweet out of the bag and eats it. She then takes out a second sweet.

(a) If the first sweet is green, what is the probability that the second sweet is also green?

(b) If the first sweet is not red, what is the probability that the second sweet is red?

Working:

Answers:(a) …………………………………………..(b) …………………………………………..

(Total 4 marks)

3

4. In a school, 180 pupils are asked which is their favourite outdoor sport in winter. The pie chart shows the result of the survey. The diagram is not accurately drawn.

F o o tb a ll(5 6 )

H o ck ey(4 3 )

R u g b y(3 8 )

N e tb a ll(3 4 )

O th e r

(a) Calculate the angle of the sector representing rugby.

(b) Estimate the probability that a pupil’s favourite outdoor sport in winter will be hockey.

Working:

Answers:

(a) …………………………………………..(b) …………………………………………..

(Total 4 marks)

4

5. On a certain game show, contestants spin a wheel to win a prize, as shown in the diagram. The larger angles are 40° (the shaded sectors), and the smaller angles are 20°.

N o P rize

N o P rize

N o P rizeN o P rize

N o P riz e

N o P rizeG H

G H

W D

W D

W D

C A R

Find the probability that a contestant

(a) will not win a prize;

(b) will win a holiday in Greece (GH);

(c) will win a washer/dryer (WD), given that he knows that he has won a prize;

5

(d) will win a holiday in Greece or a washer/dryer.

Working:

Answers:

(a) …………………………………………..(b) …………………………………………..(c) …………………………………………..(d) …………………………………………..

(Total 4 marks)

6. Of a group of five students, two will be selected to visit the United Nations. The five students are John, Maria, Raul, Henri and Susan.

(a) With the aid of a tree diagram or a table of outcomes, find the number of different possible combinations of students that could go to the United Nations.

6

(b) Find the probability that both Maria and Susan will go on the trip.

Working:

Answers:

(a) …………………………………………..(b) …………………………………………..

(Total 4 marks)

7. A group of 25 females were asked how many children they each had. The results are shown in the histogram below.

1 09876543210

0 1 2 3 4N u m b e r o f C h ild ren

F req u e n cy

N u m b e r o f C h ild ren p e r F e m ale

(a) Show that the mean number of children per female is 1.4.(2)

(b) Show clearly that the standard deviation for this data is approximately 1.06.(3)

7

(c) Another group of 25 females was surveyed and it was found that the mean number of children per female was 2.4 and the standard deviation was 2. Use the results from parts (a) and (b) to describe the differences between the number of children the two groups of females have.

(2)

(d) A female is selected at random from the first group. What is the probability that she has more than two children?

(2)

(e) Two females are selected at random from the first group. What is the probability that

(i) both females have more than two children?(2)

(ii) only one of the females has more than two children?(3)

(iii) the second female selected has two children given that the first female selected had no children?

(1)(Total 15 marks)

8. Nene and Deka both play netball. The probability that Nene will score a goal on her first attempt is 0.75. The probability that Deka will score a goal on her first attempt is 0.82.

Calculate the probability that

(a) Nene and Deka will both score a goal on their first attempts;

8

(b) neither Nene nor Deka will score a goal on their first attempts.

Working:

Answers:

(a) …………………………………………..(b) …………………………………………..

(Total 4 marks)

9. In a club with 60 members, everyone attends either on Tuesday for Drama (D) or on Thursday for Sports (S) or on both days for Drama and Sports.

One week it is found that 48 members attend for Drama and 44 members attend for Sports and x members attend for both Drama and Sports.

(a) (i) Draw and label fully a Venn diagram to illustrate this information.(3)

(ii) Find the number of members who attend for both Drama and Sports.(2)

(iii) Describe, in words, the set represented by (D S)'.(2)

(iv) What is the probability that a member selected at random attends for Drama only or Sports only?

(3)

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The club has 28 female members, 8 of whom attend for both Drama and Sports.

(b) What is the probability that a member of the club selected at random

(i) is female and attends for Drama only or Sports only?(2)

(ii) is male and attends for both Drama and Sports?(2)

(Total 14 marks)

10. Two jars contain a number of coloured balls as indicated in the diagrams below.

2 B lack3 W h ite

2 B lack1 W h ite

Jar One Jar Two

Two experiments are carried out.

First Experiment: A jar is first chosen at random and then a ball is drawn from that jar.

(a) Draw, and label fully, a tree diagram to show all possible outcomes of this experiment.(2)

(b) What is the probability that a white ball is drawn?(3)

Second Experiment: The ball drawn in the first experiment is not replaced. A second ball is then drawn from the same jar.

(c) What is the probability that both balls are white?(2)

(Total 7 marks)

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11. Members of a certain club are required to register for one of three games, billiards, snooker or darts.

The number of club members of each gender choosing each game in a particular year is shown in the table below.

Billiards Snooker Darts

Male 39 16 8

Female 21 14 17

(a) Use a 2 (Chi-squared) test at the 5% significance level to test whether choice of games

is independent of gender. State clearly the null and alternative hypotheses tested, the expected values, and the number of degrees of freedom used.

(13)

The following year the choice of games was widened and the figures for that year are as follows:

Billiards Snooker Darts Fencing

Male 4 15 8 10

Female 10 21 17 37

(b) If the 2 test were applied to this new set of data,

(i) why would it be necessary to combine billiards with another game?

(ii) which other game would you combine with billiards and why?(2)

A club member is to be selected at random.

(c) What is the probability that the club member selected is a

(i) female who chose billiards or snooker?

(ii) male or female who chose darts or fencing?(2)

(Total 17 marks)

11

12. A bag contains two red sweets and three green sweets. Jacques takes one sweet from the bag, notes its colour, then eats it. He then takes another sweet from the bag.

Complete the tree diagram below to show all probabilities.

R e d

G reen

R ed

G reen

R ed

G reen

25

24

Working:

(Total 4 marks)

12

13. In a survey of 52 students it was found that 30 study Spanish and 15 have computers. Seven of the students who study Spanish also have computers.

(a) Copy and complete this table.

StudySpanish

Do not studySpanish Total

Have computers

Do not have computers

Total 52(3)

(b) Draw and label fully a Venn diagram to illustrate this information. Use U to represent the set of all students surveyed, S the set of students who study Spanish and C the set of students who have computers.

(2)

(c) Describe, in words, the set represented by C S .(2)

(d) Find n(C S ).(1)

13

A student is selected at random to attend a computer workshop given in Spanish.

(e) What is the probability that the student

(i) has a computer and studies Spanish?(2)

(ii) as a computer but does not study Spanish? (2)

(iii) as a computer if he/she studies Spanish?(2)

(Total 14 marks)

14. The sets A, B and C are subsets of U. They are defined as follows:

U = {positive integers less than 16}

A = {prime numbers}

B = {factors of 36}

C = {multiples of 4}

(a) List the elements (if any) of the following:

(i) A;

(ii) B;

(iii) C;

(iv) A B C.(4)

(b) (i) Draw a Venn diagram showing the relationship between the sets U, A, B and C.

(ii) Write the elements of sets U, A, B and C in the appropriate places on the Venn diagram.

(4)

14

(c) From the Venn diagram, list the elements of each of the following

(i) A (B C);

(ii) (A B);

(iii) (A B) C.(3)

(d) Find the probability that a number chosen at random from the universal set U will be

(i) a prime number;

(ii) a prime number, but not a factor of 36;

(iii) a factor of 36 or a multiple of 4, but not a prime number;

(iv) a prime number, given that it is a factor of 36.(6)

(Total 17 marks)

15. A teacher has a box containing six type A calculators and four type B calculators.

The probability that a type A calculator is faulty is 0.1 and the probability that a type B calculator is faulty is 0.12.

(a) Complete the tree diagram given below, showing all the probabilities.

FA U LT Y

N O T FA U LT Y

FA U LT Y

N O T FA U LT Y

0 .1

ty p e A

ty p e B

0 .6

0 .4

15

(b) A calculator is selected at random from the box. Find the probability that the calculator is

(i) a faulty type A;

(ii) not faulty.

Working:

Answers:

(b) (i) ……………………………………...(ii) ……………………………………...

(Total 4 marks)

16. Let F be the set of all families that have exactly 2 children.

(a) Assuming P(boy) = P(girl), copy and complete the following tree diagram, for families with 2 children.

B o y

G irl

B o y

G irlB o y

G irl

12

(2)

16

(b) What is the probability that a family chosen at random from F has exactly

(i) 2 boys?

(ii) 2 boys, if it is known that the first child is a boy?

(iii) 2 boys, if it is known that there is a boy in the family?(3)

(Total 5 marks)

17. The Venn diagram below shows the number of students studying Science (S), Mathematics (M) and History (H) out of a group of 20 college students. Some of the students do not study any of these subjects, 8 study Science, 10 study Mathematics and 9 study History.

U

S M

H

41

21 3

3

A

(a) (i) How many students belong to the region labelled A?

(ii) Describe in words the region labelled A.

(iii) How many students do not study any of the three subjects?(5)

(b) Draw a sketch of the Venn diagram above and shade the region which represents S H.(1)

(c) Calculate n(S H).(2)

17

This group of students is to compete in an annual quiz evening which tests knowledge of Mathematics, Science and History. The names of the twenty students are written on pieces of paper and then put into a bag.

(d) One name is randomly selected from the bag. Calculate the probability that the student selected studies

(i) all three subjects;

(ii) History or Science.(2)

(e) A team of two students is to be randomly selected to compete in the quiz evening. The first student selected will be the captain of the team. Calculate the probability that

(i) the captain studies all three subjects and the other team member does not study any of the three subjects;

(ii) one student studies Science only and the other student studies History only;

(iii) the second student selected studies History, given that the captain studies History and Mathematics.

(5)(Total 15 marks)

18. Heinrik rolls two 6-sided dice at the same time. One die has three red sides and three black sides. The other die has the sides numbered from 1 to 6. By means of a tree diagram, table of outcomes or otherwise, answer each of the following questions.

(a) How many different possible combinations can he roll?

(b) What is the probability that he will roll a red and an even number?

(c) What is the probability that he will roll a red or black and a 5?

18

(d) What is the probability that he will roll a number less than 3?

Working:

Answers:(a) ..................................................................(b) ..................................................................(c) ..................................................................(d) ..................................................................

(Total 8 marks)

19. Fifty students at Layton High School recorded how much money each student in their class spent on video rentals this month (to the nearest dollar). The results are shown in the frequency table below:

Class interval in $ Boundaries in $ Frequency

1–10 0.50–10.50 10

11–20 10.50–20.50 20

21–30 20.50–30.50 10

31–40 30.50–40.50 0

41–50 40.50–50.50 4

51–60 50.50–60.50 2

61–70 60.50–70.50 4

(a) On graph paper using a scale of 2 cm to represent each interval ($10.00) on the horizontal axis and 1 cm to represent 5 people on the vertical axis, draw and clearly label a frequency histogram which displays the above information.

(5)

19

(b) Answer the following questions:

(i) Which class is the modal class?

(ii) In which class is the median?(2)

(c) Assuming these students spend the same amount on videos each month, find the probability that next month a student will spend an amount in the class interval:

(i) From $21 to $30 inclusive on video rentals.

(ii) $30 or less on video rentals.

(iii) From $41 to $60 on video rentals, given that they spent more than $20 on video rentals.

(iv) Not more than $60 on video rentals, given that they spent over $10 on video rentals.

(6)(Total 13 marks)

20

20. Today Philip intends to go walking. The probability of good weather (G) is 43

. If the weather is

good, the probability he will go walking (W) is 2017

. If the weather forecast is not good (NG)

the probability he will go walking is 51

.

(a) Complete the probability tree diagram to illustrate this information.

34

G

N G

N W

N W

W

W

1 72 0

(b) What is the probability that Philip will go walking?

Working:

Answer:

(b) ..................................................................

(Total 8 marks)

21. On a particular day 100 children are asked to make a note of what they drank that day.

21

They are given three choices: water (W), coffee (C) or fruit juice (F)

1 child drank only water.6 children drank only coffee.8 children drank only fruit juice.5 children drank all three.7 children drank water and coffee only.53 children drank coffee and fruit juice only.18 children drank water and fruit juice only.

(a) Represent the above information on a Venn Diagram.(4)

(b) How many children drank none of the above?(2)

(c) A child is chosen at random. Find the probability that the child drank

(i) coffee;

(ii) water or fruit juice but not coffee;

(iii) no fruit juice, given that the child did drink water.(4)

(d) Two children are chosen at random. Find the probability that both children drank all three choices.

(3)(Total 13 marks)

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22. Two identical dice have sides numbered one to six. The dice are weighted. All the numbers except the four have equal probability of appearing on top. The four is three times as likely as each of the other numbers to appear on top.

The tree diagram below shows some of the probabilities.

a8

18

18

b8

181

8

o n e

tw o

th ree

fo u r

fiv e

six

(a) Find the values of a and b in the diagram.

(b) Both dice are thrown. Calculate the probability that two fours appear on top.

23

(c) One of the dice is thrown once. The result is not a two or a three. What is the probability that it is a six?

Working:

Answers:

(a) ..................................................................(b) ..................................................................(c) ..................................................................

(Total 8 marks)

23. There are two biscuit tins on a shelf. The red tin contains three chocolate biscuits and seven plain biscuits. The blue tin contains one chocolate biscuit and nine plain biscuits.

(a) A child reaches into the red tin and randomly selects a biscuit. The child returns that biscuit to the tin, shakes the tin, and then selects another biscuit.

Find the probability that

(i) both biscuits chosen are chocolate.(2)

(ii) one of the biscuits is plain and the other biscuit is chocolate.(3)

24

(b) A second child chooses a biscuit from the blue tin. The child eats the biscuit and chooses another one from the blue tin. The tree diagram below represents the possible outcomes for this event.

11 0

91 0

ab

C

C

C

P

P

P

(i) Write down the values of a and b.(2)

(ii) Find the probability that both biscuits are chocolate.(1)

(iii) What is the probability that at least one of the biscuits is chocolate?(3)

(c) Suppose that before the two children arrived, their brother randomly selected one of the biscuit tins and took out one biscuit.

Calculate the probability that this biscuit was chocolate.(4)

(Total 15 marks)

25

24. The probability, p, that James gets up before 07.00 is 0.95.If James gets up before 07.00, the probability, t, that he arrives at school on time is 0.98. If James gets up later than 07.00, the probability that he arrives at school on time is 0.55.The above information is represented by the following tree diagram.

0 .9 5

0 .5 5

0 .9 8

n o t p

n o t t

n o t t

p

t

t

(a) Complete the tree diagram.

(b) Calculate the probability that James gets up before 07.00 and is on time for school.

(c) Calculate the probability that James does not arrive at school on time.

Working:

Answers:

(b) ..................................................................(c) ..................................................................

(Total 8 marks)

26

25. A school jazz band contains three different musical instruments ― saxophone (S), clarinet (C) and drums (D). Students in the band are able to play one, two or three different instruments.In a class of 40 IB students, 25 belong to the jazz band. Out of these 25

3 can play all three instruments5 can play the saxophone and clarinet only5 can play at least the clarinet and drums7 can play at least the saxophone and drums16 can play the saxophone12 can play the clarinet.

(a) Draw a Venn Diagram and clearly indicate the numbers in each region.(5)

(b) Show that the number of students who play the drums only is 5.(2)

(c) Find the probability that a student chosen at random from the IB class plays only the saxophone.

(2)

(d) Find the probability that a student chosen at random from the IB class plays either the clarinet or drums or both.

(2)

(e) Given that a student plays the saxophone, find the probability that he also plays the clarinet.

(3)(Total 14 marks)

27

26. The following histogram shows the house prices in thousands of Australian dollars (AUD) of a random sample of houses in a certain town in Australia.

4 5

4 0

3 5

3 0

2 5

2 0

1 5

1 0

5

0 0 6 0 1 2 0 1 8 0 2 4 0 3 0 0 T h o u sa n d s o fd o lla rs

F req u en cy

(a) How many houses are there in the sample?

(b) Write down the modal group for house prices.

(c) Find the probability of choosing a house at random which costs less than 60 000 AUD or more than 240 000 AUD.

(d) Given that a house costs more than 120 000 AUD, find the probability that it costs between 180 000 and 240 000 AUD.

Wo rk in g :

A n sw ers:

(a ) ..... .. .. .. .... ........... .. .. .. .. .. .. .............. .

(b ) ......... .. . ... .. .. .. .. .. ..... .. .. .. .. .. .. .. .. .. .. ...

(c ) ... .. .. .. .. .... ........... .. .. .. .. .. .. .............. .

(d ) ......... .. . ... .. .. .. .. .. ..... .. .. .. .. .. .. .. .. .. .. ...(Total 6 marks)

28

27. Sandra is attempting an exam question. She has to choose two correct statements from a list of five. Below is a tree diagram showing Sandra’s possible choices. One of the probability values is missing.

co rrec t

co rrec t

in co rrec t

co rrec tin co rrec t

in co rrec t

2–5

35–

34–

24–

—

24–

(a) Fill in the missing probability value on the diagram.

29

(b) (i) If Sandra makes two guesses, what is the probability that she will get only one of them correct?

(ii) Sandra definitely knows the first correct statement but has to guess the second. What is the probability that she will answer both correctly?

Wo rkin g:

A nsw ers:

(b ) (i) ............ .. .. .. .. .. ........... .. .. .. .. .. .. .. ..

(ii) ..... .. .. .. .. .. .. .. .. .. ......... .. .. .. .. .. .. .. ..(Total 8 marks)

30

28. The data in the table below refers to a sample of 60 randomly chosen plants.

Growth rate Classification by environment dark light shady total

high 3 8 14 25

low 8 9 18 35

total 11 17 32 60

31

(a) (i) Find the probability of a plant being in a shady environment.

32

(ii) Find the probability of a plant having a low growth rate and being in a dark environment.

33

(iii) Find the probability of a plant not being in a dark environment.(5)

34

(b) A plant is chosen at random from the above group.

Find the probability that the chosen plant has

(i) a high growth rate or is in a dark environment, but not both

(ii) a light environment, given that it has a high growth rate.(4)

35

(c) The 60 plants in the above group were then classified according to leaf type. It was found that 15 of the plants had type A leaves, 37 had type B leaves and 8 had type C leaves.

Two plants were randomly selected from this group. Find the probability that

(i) both plants had type C leaves

(ii) neither of the plants had type B leaves.(5)

(Total 14 marks)

36

29. Children in a class of 30 students are asked whether they can swim (S) or ride a bicycle (B).

There are 12 girls in the class. 8 girls can swim, 6 girls can ride a bicycle and 4 girls can do both.

16 boys can swim, 13 boys can ride a bicycle and 12 boys can do both. This information is represented in a Venn diagram.

U B o y s G irls

S

B

41 2

b

a

2

37

(a) Find the values of a and b.(2)

38

(b) Calculate the number of students who can do neither.(2)

39

(c) Write down the probability that a student chosen at random can swim.(2)

40

(d) Given that the student can ride a bicycle, write down the probability that the student is a girl.

(2)(Total 8 marks)

41

30. In a group of fifteen students, three names begin with the letter B and four begin with a G. The remaining eight names begin with A, C, D, E, F, H, I and J respectively.

The 15 names are placed in a box. The box is shaken and two names are drawn out.

Find the probability that

(a) both names begin with any letter except G or B;

(b) both names begin with the same letter;

(c) both names begin with the letter H.

A n sw ers:

Work in g:

(a ) ......... .. .. ......... . .. ... .. .. .. .. .. .. ....... .. ....

(b ) ............. .. .. ....... .. .. .. .. .. .. .. .. .. ... .. .. ....

(c ) ... .. .. .. .. .. .. .. ... .. .. .. .. .. .. .................. .. .(Total 6 marks)

42

31. Let U be the set of all positive integers from 1 to 21 inclusive.

A, B and C are subsets of U such that:

A contains all the positive integers that are factors of 21,B is the set of multiples of 7 contained in U,C is the set of odd numbers contained in U.

(a) List all the members of set A.(2)

(b) Write down all the members of

(i) A ∪ B;

(ii) C′ ∪B.(4)

(c) Find the probability that a member chosen at random from A is also a member ofA ∪ B ∪ C.

(2)(Total 8 marks)

43

32. The table below shows the number of words in the extended essays of an IB class.

Number ofwords 3200w3400 3400w3600 3600w3800 3800w4000 4000w4200

Frequency 2 5 8 17 3

(a) Draw a histogram on the grid below for the data in this table.

5

1 0

1 5

2 0

3 0 0 0 3 2 0 0 3 4 0 0 3 6 0 0 3 8 0 0 4 0 0 0 4 2 0 0 4 4 0 0

Freq

uenc

y

N u m b e r o f w o rd s

0

(3)

(b) Write down the modal group.(1)

The maximum word count is 4000 words.

(c) Write down the probability that a student chosen at random is on or over the word count.(2)

Wo rking :

A n sw ers:

(c ) ............ ..... .. .. .. .. ............... .. .. .. .. .. .

(b ) ......... .. . ... .. .. .. .. .. ..... .. .. .. .. .. .. .. .. .. .. .

(Total 6 marks)

44

45

33. The following histogram shows the weights of a number of frozen chickens in a supermarket. The weights are grouped such that 1 weight 2, 2 weight, 3 and so on.

5 5

5 0

4 5

4 0

3 5

3 0

2 5

2 0

1 5

1 0

5

06543210

n u m b e r o fch ick en s

w e ig h t (k g )

(a) On the graph above, draw in the frequency polygon.(2)

(b) Find the total number of chickens.(1)

(c) Write down the modal group.(1)

Gabriel chooses a chicken at random.

(d) Find the probability that this chicken weighs less than 4 kg.(2)

46

A n sw ers:

Work in g:

(a ) ..... .. ......... .. .. ... .. ........... .. .. .. .. ..... . ...

(b ) ..... .. .. .. ......... .. ................ .. .. ........ .. .

(c ) ............. .. .. ......... .. .. .. .. .. .. .. .. ... .. .. ....(Total 6 marks)

34. A group of 50 students completed a questionnaire for a Mathematical Studies project. The following data was collected.

18 students own a digital camera (D)15 students own an ipod (I)26 students own a cell phone (C)1 student owns all three items5 students own a digital camera and an ipod but not a cell phone2 students own a cell phone and an ipod but not a digital camera3 students own a cell phone and a digital camera but not an ipod

(a) Represent this information on a Venn diagram.(4)

(b) Calculate the number of students who own none of the items mentioned above.(2)

(c) If a student is chosen at random, write down the probability that the student owns a digital camera only.

(1)

(d) If two students are chosen at random, calculate the probability that they both own a cell phone only.

(3)

(e) If a student owns an ipod, write down the probability that the student also owns a digital camera.

(2)(Total 12 marks)

47

35. Jim drives to work each day through two sets of traffic lights.

The probability of the first set of traffic lights being red is 0.65.

If the first set is red then the probability that the next set of traffic lights is red is 0.46.

If the first set is not red, the probability that the next set is red is 0.72.

0 .6 5

n o t

re d

re d

re d

n o tre d

re d

n o tre d

(a) Complete the tree diagram above.

(b) Calculate the probability that the second set of traffic lights is red.

Wo rk ing :

A n sw er:

(b ) .... ... .. .. ... .. ... .. .. .. .... .. .... .. .... .. .... .. ...

(Total 8 marks)

48

36. Amos travels to school either by car or by bicycle. The probability of being late for school is

101

if he travels by car and 51

if he travels by bicycle. On any particular day he is equally likely to travel by car or by bicycle.

(a) Draw a probability tree diagram to illustrate this information.(4)

(b) Find the probability that

(i) Amos will travel by car and be late.(2)

(ii) Amos will be late for school.(3)

(c) Given that Amos is late for school, find the probability that he travelled by bicycle.(3)

(Total 12 marks)

37. Neil has three dogs. Two are brown and one is grey. When he feeds the dogs, Neil uses three bowls and gives them out randomly. There are two red bowls and one yellow bowl. This information is shown on the tree diagram below.

23

13

23

23

13

13

B ro w n

G rey

R ed

Ye llo w

R ed

Ye llo w

(a) One of the dogs is chosen at random.

(i) Find P (the dog is grey and has the yellow bowl).

(ii) Find P (the dog does not get the yellow bowl).(3)

49

(b) Neil often takes the dogs to the park after they have eaten. He has noticed that the grey dog plays with a stick for a quarter of the time and both brown dogs play with sticks for half of the time. This information is shown on the tree diagram below.

23

13

B ro w n

G re y

S tic k

N ostick

S tic k

N ostick

(i) Copy the tree diagram and add the four missing probability values on the branches that refer to playing with a stick.

During a trip to the park, one of the dogs is chosen at random.

(ii) Find P (the dog is grey or is playing with a stick, but not both).

(iii) Find P (the dog is grey given that the dog is playing with a stick).

(iv) Find P (the dog is grey and was fed from the yellow bowl and is notplaying with a stick).

(9)(Total 12 marks)

50

38. When Geraldine travels to work she can travel either by car (C), bus (B) or train (T). She travels by car on one day in five. She uses the bus 50 of the time. The probabilities of her being late (L) when travelling by car, bus or train are 0.05, 0.12 and 0.08 respectively.

(a) Copy the tree diagram below and fill in all the probabilities, where NL represents not late, to represent this information.

(5)

L

L

LN L

N L

N L

C

B

T

0 .2

(b) Find the probability that Geraldine travels by bus and is late.(1)

(c) Find the probability that Geraldine is late.(3)

(d) Find the probability that Geraldine travelled by train, given that she is late.(3)

(Total 12 marks)

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39. When Andy plays tennis, 65 of his first serves go into the correct area of the court.If the first serve goes into the correct area, his chance of winning the point is 90.If his first serve does not go into the correct area, Andy is allowed a second serve and, of these, 80 go into the correct area.If the second serve goes into the correct area, his chance of winning the point is 60.If neither serve goes into the correct area, Andy loses the point.

(a) Complete the tree diagram below.

0 .6 5

0 .3 5

0 .2

.....

0 .8

0 .1

.....

L o se

W i n

0 .6

0 .4

W i n

L o se

L o se

S e co n d se rv e in

S e co n d serv e o u t

F irs t se rv e o u t

F irs t se rv e in

(2)

(b) Find the probability that Andy loses the point.(4)

Wo rk ing :

A n sw er:

(b ) ....... .. .. . .. .. ..... .. .... .. .... .. .. .. .. .. .. .. .. ...

(Total 6 marks)

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40. Claire and Kate both wish to go to the cinema but one of them has to stay at home to baby-sit.

The probability that Kate goes to the cinema is 0.2. If Kate does not go Claire goes.If Kate goes to the cinema the probability that she is late home is 0.3.If Claire goes to the cinema the probability that she is late home is 0.6.

(a) Copy and complete the probability tree diagram below.

0 .2

0 .3

L a te

L a te

N o t L a te

N o t L a te

K a te

C la ire. . .

. . .

. . .

. . .

(3)

(b) Calculate the probability that

(i) Kate goes to the cinema and is not late;(2)

(ii) the person who goes to the cinema arrives home late.(3)

(Total 8 marks)

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41. The table below shows the number of left and right handed tennis players in a sample of 50 males and females.

Left handed Right handed Total

Male 3 29 32

Female 2 16 18

Total 5 45 50

If a tennis player was selected at random from the group, find the probability that the player is

(a) male and left handed;

(b) right handed;

(c) right handed, given that the player selected is female.

Working:

Answers:(a) …………………………………………..(b) …………………………………………..(c) …………………………………………..

(Total 4 marks)

42. It is known that 5% of all AA batteries made by Power Manufacturers are defective. AA batteries are sold in packs of 4.

Find the probability that a pack of 4 has

(a) exactly two defective batteries;(3)

(b) at least one defective battery.(2)

(Total 5 marks)

43. Events A and B have probabilities P(A) = 0.4, P (B) = 0.65, and P(A B) = 0.85.

54

(a) Calculate P(A B).

(b) State with a reason whether events A and B are independent.

(c) State with a reason whether events A and B are mutually exclusive.

Workin g :

A n sw ers:(a ) ....................... ..... .. .. .. .. .. .. .. ... .. .. ..

(b ) ..... .. .. .. .. .. .. .. .. ... .......... .............. .. .

...... .. .. .. ....... .. .. .. .. .. .. .. .. .. .. ..... .. .. .. .

...... .. .. .. ....... .. .. .. .. .. .. .. .. .. .. ..... .. .. .. .

(c ) ....................... ..... .. .. .. .. .. .. .. ... .. .. ..

...... .. .. .. ....... .. .. .. .. .. .. .. .. .. .. ..... .. .. .. .

...... .. .. .. ....... .. .. .. .. .. .. .. .. .. .. ..... .. .. .. .

(Total 6 marks)

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44. There are 49 mice in a pet shop.30 mice are white.27 mice are male.18 mice have short tails.8 mice are white and have short tails.11 mice are male and have short tails.7 mice are male but neither white nor short-tailed.5 mice have all three characteristics and2 have none.

Copy the diagram below to your examination script.

W re p re se n ts w h ite m ice .M re p re se n ts m a le m ic e .S re p rese n ts sh o rt- ta ile d m ic e .

S W

M

U

3

5

(a) Complete the diagram, using the information given in the question.(4)

(b) Find (i) n(M W)

(ii) n(M′ S)(3)

Two mice are chosen without replacement.

(c) Find P (both mice are short-tailed).(2)

(Total 9 marks)

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45. A survey of 400 people is carried out by a market research organization in two different cities, Buenos Aires and Montevideo. The people are asked which brand of cereal they prefer out of Chocos, Zucos or Fruti. The table below summarizes their responses.

Chocos Zucos Fruti Total

Buenos Aires 43 85 62 190

Montevideo 57 35 118 210

Total 100 120 180 400

(a) One person is chosen at random from those surveyed. Find the probability that this person

(i) does not prefer Zucos;

(ii) prefers Chocos, given that they live in Montevideo.(4)

(b) Two people are chosen at random from those surveyed. Find the probability that they both prefer Fruti.

(3)

The market research organization tests the survey data to determine whether the brand of cereal preferred is associated with a city. A chi-squared test at the 5 level of significance is performed.

(c) State the null hypothesis.(1)

(d) State the number of degrees of freedom.(1)

(e) Show that the expected frequency for the number of people who live in Montevideo and prefer Zucos is 63.

(2)

(f) Write down the chi-squared statistic for this data.(2)

(g) State whether the market research organization would accept the null hypothesis. Clearly justify your answer.

(2)(Total 15 marks)

57