Statistics and Probability

160.
The bar chart shows the marks (%) obtained by some students in a test.
Percentage Marks Obtained by Students in a Test 0 2 4 6 8 10 Number of students 30 40 50 60 70 80 Marks in %
(a) Find the modal mark.
(a) ......................... % [1]
(b) Find the probability that a student scores:
(i) More than 60%,
(b)(i) ......................... [1]
(ii) 40% or 70%.
(b)(ii) ......................... [1]
Paper 2
161.
Bags A and B contain balls of the same shape and size.
Bag A contains 4 red balls and 2 yellow balls.
Bag B contains 2 red balls and 3 yellow balls.
(a) A ball is drawn at random from each bag. Find the probability that both balls are of the same colour.
(a) ......................... [3]
(b) Two balls are drawn at random from each bag, one at a time.
Find the probability that:
(i) The two balls drawn from bag A are red.
(b)(i) ......................... [3]
(ii) All the four balls drawn are red.
(b)(ii) ......................... [1]
Paper 2
162.
In one school, the probability that students pass grade 9 is 3 5 .
A student who passes grade 9 progresses to grade 10, or else repeats grade 9.
The probability that those who pass progress in the same school is 2 3 .
The probability that those who fail repeat in the same school is 1 4 .
The tree diagram shows this information.
Pass 3 5 Fail Progresses in the same school 2 3 Progresses in a different school Repeats in the same school 1 4 Repeats in different school
(a) Complete the tree diagram.
(a) See diagram [2]
(b) If the school has 300 grade 9 students, calculate the number of students who progress in the same school.
(b) ......................... [2]
Paper 2
163.
The pie chart shows information about 60 fruit in an orchard.
apple pea peach apricot grape
(a) Find the modal fruit tree.
(a) ......................... [1]
(b) Find the number of apple trees in the orchard.
(b) ......................... [2]
Paper 2
164.
The range of the distribution 4, p, 7, 12 is 9.
Find all possible values of p.
p = ......................... [3]
Paper 2
165.
The cumulative frequency curve represents the distribution of heights of 120 seedlings.
Height (cm) Cumulative frequency 0 40 80 120 31 32 33
Use the curve to estimate:
(a) The median height,
(a) ......................... cm [1]
(b) The upper quartile range,
(b) ......................... cm [1]
(c) The 90th percentile.
(c) ......................... cm [2]
Paper 2
166.
The probability that it rains (R) on the first day is 3 4 .
The probability that it does not rain (NR) on the second day is 3 5 .
(a) Complete the tree diagram.
First day Second day R 3 4 NR R NR 3 5 R NR 3 5
(a) See diagram [1]
(b) Find the probability that:
(i) It rains on both days,
(b)(i) ......................... [1]
(ii) It does not rain on exactly one day.
(b)(ii) ......................... [3]
Paper 2
167.
The probabilities that three football teams, A, B and C, win their next game are 1 2 , 1 4 and 1 5 respectively.
Find the probability that:
(a) None of the three teams win.
(a) ......................... [2]
(b) One of the three teams wins.
(b) ......................... [3]
Paper 2
168.
(a) The numbers show goals scored in six football games:
8, x, 1, 2, 11, 1
Given that the median is 3 1 2 , find the value of x.
(a) x = ......................... [2]
(b) The mean of eight numbers is 3. The mean of a different set of twelve numbers is y. Given that the mean of these twenty numbers is 9, calculate the value of y.
(b) y = ......................... [3]
Paper 2
169.
The table shows the length, l mm, of the 40 leaves.
Length of a leaf (l mm) 0 < l ≤ 10 10 < l ≤ 20 20 < l ≤ 60 60 < l ≤ 80
Frequency 6 20 4 10
(a) Find the modal class.
(a) ......................... [1]
(b) Calculate an estimate of the mean.
(b) ......................... mm [4]
(c) On a histogram, the height of the interval 60 < l ≤ 80 is 2.5 cm. Calculate the height of the interval 10 < l ≤ 20.
(c) ......................... cm [2]
Paper 2
170.
A report indicates that the number of arrests of criminals has doubled from months May to July as shown by the graph.
20 50 100 Number of criminal arrests May June Month July
Explain why this graph is misleading.
......................... [1]
Paper 2
171.
A box contains 20 coloured marbles; 9 blue, 6 purple and 5 yellow.
(a) A marble is taken at random from the box. Find the probability that the marble is blue.
(a) ......................... [2]
(b) Two marbles are taken at random without replacing.
(i) Show that the probability of taking out the two purple marbles is 3 38 .
(b)(i) ......................... [2]
(ii) Calculate the probability that both marbles are of a different colour.
(b)(ii) ......................... [3]
Paper 2
172.
Given the distribution 12, 10, 8, 15, 18, 6, 8.
(a)(i) Find the lower quartile.
(a)(i) ......................... [1]
(ii) Find the inter-quartile range.
(a)(ii) ......................... [2]
(b) Calculate the mean.
(b) ......................... [2]
Paper 2
105.
The frequency table shows the number of hectares (ha) owned by 160 farmers.
ha 0 < ha ≤ 10 10 < ha ≤ 40 40 < ha ≤ 60 60 < ha ≤ 90 90 < ha ≤ 100 100 < ha ≤ 120
f 3 39 43 55 11 9

(a) The height of the bar representing the class 0 < ha ≤ 10 is 0.3.
Find the height of the bar representing the class 10 < ha ≤ 40.
......................... [2]
(b) One farmer is chosen at random.
Find the probability that the farmer owns more than 90 hectares.
......................... [2]
(c) Calculate an estimate of the mean number of hectares owned.
......................... hectares [4]
(d) Complete the cumulative frequency table shown.
Hectares (ha) ha ≤ 10 ha ≤ 40 ha ≤ 60 ha ≤ 90 ha ≤ 100 ha ≤ 120
Cumulative
frequency		 3 85 160

(e) On the grid, draw a cumulative frequency curve.
0 10 20 30 40 50 60 70 80 90 100 110 120 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 Number of hectares Cumulative Frequency
(f) Use your cumulative frequency curve to find:
(i) the median,
......................... hectares [1]
(ii) the interquartile range,
......................... hectares [2]
(iii) the 90th percentile,
......................... hectares [2]
(iv) the number of farmers who have more than 40 hectares.
......................... [2]
Paper 4
106.
The numbers 0, 0, 1, 1, 1, 2, p, 9, 10, 13, q, 16 are in order.
Their mean is 6 and their median is 3.5.
(a) State the mode.
Mode = ......................... [1]
(b) Find the range.
Range = ......................... [1]
(c) Find the value of:
(i) p,
p = ......................... [1]
(ii) q.
q = ......................... [2]
(d) Find the probability that a number chosen at random is 0 or 1.
......................... [1]
(e) One number is chosen and not replaced.
A second number is then chosen.
Calculate the probability that the two numbers are both 0 or both 1.
......................... [3]
Paper 4
107.
The diagram shows the results of a survey on the favourite games played by a group of students.
NOT TO SCALE 30° 75° Khati Boleke Cheko Mantloane
(a) Find the probability that a student chosen at random:
(i) does not play Boleke,
......................... [2]
(ii) plays Cheko.
......................... [2]
(b) A total of 10 students play Boleke.
(i) Find the number of students in the survey.
......................... [2]
(ii) Two students are selected at random, one after the other.
Calculate the probability that the first student plays Khati and the second plays Mantloane.
......................... [3]
Paper 4
108.
The histogram shows the distribution of masses (m grams) of packets of sweets in a box.
0 10 20 30 40 50 60 70 80 90 100 0 2 4 6 8 10 Mass (grams) Frequency density
(a) There are 100 packets in the mass group 50 < m ≤ 60.
Calculate the total number of packets of sweets in the box.
......................... [3]
(b) Show that approximately 15.4% of the packets are in the mass group 30 < m ≤ 50.
......................... [2]
(c) Calculate an estimate of the mean of all the packets.
......................... [4]
(d) Estimate the number of packets with a mass of 78 < m ≤ 100.
......................... [3]
Paper 4
109.
(a) The diagram shows the frequency distribution of marks obtained by 21 students in a test.
3 students scored 10 marks.
Mark Frequency KEY = 2 students 4 5 7 8 9 10
(i) Complete the diagram.
See diagram [1]
(ii) What is the name given to this kind of diagram?
......................... [1]
(iii) Calculate the probability that a student chosen at random from the class obtained a mark lower than the mode.
......................... [2]
(iv) Calculate the mean mark.
......................... [4]
(c) In another class of 80 students, marks were presented in a table as shown.
Mark 1-10 11-20 21-30 31-40 41-50 51-60 61-70 71-80 81-90 91-100
frequency 2 1 4 9 12 12 11 10 6 3

(i) A student is chosen at random from the class.
Calculate the probability that the student has obtained more than 50 marks.
......................... [2]
(ii) Complete the cumulative frequency table.
Mark ≤ 10 ≤ 20 ≤ 30 ≤ 40 ≤ 50 ≤ 60 ≤ 70 ≤ 80 ≤ 90 ≤ 100
Cumulative
frequency		 2 3 28 50 61 71 80
(iii) On the grid, draw the cumulative frequency curve.
0 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 marks C F
[3]
(iv) Use your graph to estimate:
(a) the median,
......................... [1]
(b) the interquartile range,
......................... [2]
(c) the pass mark if 2 5 of the class failed.
......................... [2]
Paper 4
110.
A traffic department has speed records on 60 cars.
The cumulative frequency curve illustrates the information.
0 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 Speed (km/h) Cumulative frequency
(a) Estimate:
(i) The median speed,
Median = ......................... km/h [1]
(ii) The inter-quartile range.
IQR = ......................... km/h [2]
(b) The speed limit on the road is 80 km/h.
A car is chosen at random, calculate the probability that this car exceeded the speed limit.
......................... [2]
(c) Use this information from the cumulative frequency curve to complete the table.
Speed (km/h) 0 < s ≤ 10 10 < s ≤ 20 20 < s ≤ 30 30 < s ≤ 40 40 < s ≤ 50
Frequency 4 4 4 4 7
Speed (km/h) 50 < s ≤ 60 60 < s ≤ 70 70 < s ≤ 80 80 < s ≤ 90 90 < s ≤ 100
frequency 4 2
See table [3]
Paper 4
111.
The table shows the time, t minutes, taken by 60 students to complete a mathematics test.
Time t, (minutes) 0 < t ≤ 10 10 < t ≤ 15 15 < t ≤ 30
frequency 17 23 20

(a) Calculate an estimate of the mean.
......................... [3]
(b) On a histogram, the frequency density for 0 < t ≤ 10 is 1.7 cm.
Calculate the frequency density for 10 < t ≤ 15.
......................... [2]
Paper 4
112.
Mpho records the growth in the height of a tree over a period of 23 weeks.
At the start, the tree has a height of 0.78 metres tall.
End of week number 1 2 3 4 5 6 7
Height of tree (m) 0.8 0.83 0.87 0.91 0.93 0.95 1.0

(a) By how much has the tree grown during week 5? Give your answer in centimetres.
......................... cm [2]
(b) In which week is the maximum growth recorded?
Week ......................... [1]
(c) From week 8 to week 23 Mpho summarises the amount of growth in the height of the tree in the table below.
Amount of growth each
week (r cm)		 1 < r ≤ 2 2 < r ≤ 2.5 2.5 < r ≤ 3 3 < r ≤ 3.5 3.5 < r ≤ 4.5 4.5 < r ≤ 5
Number of weeks 2 1 3 3 5 2

(d) (i) Using the tables in part (a) and (c), complete the table below to cover the entire period of 23 weeks.
Amount of growth
each week (r cm)		 1 < r ≤ 2 2 < r ≤ 2.5 2.5 < r ≤ 3 3 < r ≤ 3.5 3.5 < r ≤ 4.5 4.5 < r ≤ 5
Number of weeks 5 1 4 3

(ii) On the grid provided, draw the frequency polygon to represent the information.
0 2 4 6 8 1 2 3 4 5 6 w r number of weeks amount of growth (cm)
See tables and diagram [8]
Paper 4
113.
A box contains six apples and five oranges.
Mpho takes a fruit at random from the box without replacement.
He then takes a second fruit at random.
(a) Complete the tree diagram.
First fruit Second fruit 5 11 .......... Orange Apple 4 10 .......... Orange Apple .......... .......... Orange Apple

(b) Calculate the probability that Mpho takes:
(i) Two oranges,
......................... [1]
(ii) At least one apple.
......................... [2]
(c) Mpho now takes six fruits, at random from the remaining without replacement.
Find the probability that all six fruits are oranges.
......................... [1]
Paper 4
114.
The histogram shows the distance, d metres, ran by 150 students.
0 20 40 60 80 100 120 1 2 3 4 Distance (d metres) Frequency density
(a) Complete the table.
Distance (d m) 0 < d ≤ 20 20 < d ≤ 40 40 < d ≤ 50 50 < d ≤ 60 60 < d ≤ 100
frequency 30 42 29
[2]
(b) Calculate an estimate of the mean.
......................... m [4]
(c) 10% of the children ran further than y metres.
Calculate an estimate of y.
y = ......................... [3]
Paper 4
115.
(a) In a group of students 60% are female and 40% are male.
One third of the female students study Sesotho and a quarter of the male students study Sesotho.
A student is chosen at random from the group.
(i) Find the probability that the student is a female and studies Sesotho.
......................... [2]
(ii) Find the probability that the student studies Sesotho.
......................... [2]
(b) 40 students completed a puzzle.
The cumulative frequency curve shows the time in seconds taken to complete the puzzle.
0 10 20 30 40 10 20 30 40 50 60 Cumulative frequency Time in seconds
Use the graph to find:
(i) the median time taken to complete the puzzle,
Median = ......................... [2]
(ii) the inter-quartile range,
IQR = ......................... [2]
(iii) the number of students who completed the puzzle in more than 35 seconds.
......................... [2]
Paper 4
116.
The table shows the time (t seconds) taken by 100 candidates to answer a given question.
t 0 < t ≤ 20 20 < t ≤ 30 30 < t ≤ 40 40 < t ≤ 50 50 < t ≤ 60 60 < t ≤ 80
Frequency 10 10 43 22 7 8

(a) Calculate an estimate of the mean time taken. Show your working.
Mean = ......................... [4]
(b) The data is regrouped to give the following table.
t 0 < t ≤ 20 20 < t ≤ 30 30 < t ≤ 40 40 < t ≤ 50 50 < t ≤ 60 60 < t ≤ 80
Frequency 10 10 43 22 7 8
(i) Write the values of v and w.
v = .........................
w = ......................... [2]
(ii) On the grid, draw a histogram which shows the information in the table in part (b).
0 0.5 1.0 1.5 2.0 2.5 3.0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 Frequency density Time (sec)
[3]
Paper 4
117.
(a) There are 25 red marbles and x blue marbles in a box.
One marble is selected at random.
(i) Given that the probability that it is blue is 1 6 , calculate the value of x.
x = ......................... [3]
(ii) Find the number of marbles that are in the box.
......................... [1]
(b) In another box there are 27 green marbles and 113 yellow marbles.
Two marbles are selected at random without replacement.
Calculate the probability that:
(i) Both marbles are green,
......................... [2]
(ii) There is at least one yellow marble.
......................... [2]
Paper 4
118.
The table shows the masses, m grams, of 44 plant seeds.
Mass (g) 0 ≤ m < 20 20 ≤ m < 25 25 ≤ m < 35 35 ≤ m < 50 50 ≤ m < 55
frequency 8 5 11 18 2

(a) (i) Find the modal class.
......................... [1]
(ii) Find the class that contains the median.
......................... [1]
(iii) Calculate an estimate of the mean.
Mean = ......................... g [4]
(b) Draw a histogram to represent the information in the table.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 10 20 30 40 50 60 Mass (g)
See diagram [4]
Paper 4
119.
(a) The table below shows the marks scored by a group of students in a test.
Marks 5 6 7 8 9 10
frequency 2 8 5 6 4 2

Find:
(i) The mode,
Mode = ......................... [1]
(ii) The median,
Median = ......................... [2]
(iii) The mean.
Mean = ......................... [3]
(b) The table below shows the time (t minutes) taken by the students to complete their homework.

Time ( t minutes) 0 < t ≤ 10 10 < t ≤ 20 20 < t ≤ 30 30 < t ≤ 40
frequency 2 15 7 3

(i) Calculate an estimate of the mean.
Mean = ......................... min [4]
(ii) On the grid, draw a histogram to show this information.
2.0 1.5 1.0 0.5 0 10 20 30 40 50 t Frequency density Time (minutes)
[3]
Paper 4
120.
The diagram shows two sets of cards.
Set A T H A T O Set B T A U
(a) One card is chosen at random from Set A and replaced.
(i) Write down the probability that the card chosen shows the letter A.
......................... [1]
(ii) Write down the probability that the card chosen does not show the letter A.
......................... [1]
(iii) If this is carried out 200 times, write down the expected number of times the card chosen does not show the letter A.
......................... [1]
(iv) Write down the probability that the card chosen shows the letter M.
......................... [1]
(b) Two cards are chosen at random, without replacement, from Set A.
(i) Find the probability that both cards show the letter T.
......................... [2]
(ii) Find the probability that one card shows the letter T and one card shows the letter H.
......................... [3]
(c) One card is chosen at random from Set A and one is chosen at random from Set B.
Find the probability that at least one of the two cards shows the letter T.
......................... [3]
(d) A card is chosen at random, without replacement, from Set A until the letter shown is T.
Find the probability that this does not happen until the 3rd card is chosen.
Show your working.
......................... [2]
Paper 4
121.
The table shows the time, y years, that 150 men have been married for.

Time ( y years) 0 < y ≤ 10 10 < y ≤ 20 20 < y ≤ 30 30 < y ≤ 40 40 < y ≤ 50
frequency 30 50 35 25 10

(a) Calculate an estimate of the mean.
Mean = ......................... years [4]
(b) The table shows the cumulative frequencies.

Time ( y years) y ≤ 10 y ≤ 20 y ≤ 30 y ≤ 40 y ≤ 50
Cumulative
frequency		 30 80 115 140 150

(i) Use the information from the table to draw a cumulative frequency curve on the grid.
160 140 120 100 80 60 40 20 0 10 20 30 40 50 Cumulative frequency Time (years)
[3]
(ii) Use your graph to find:
(a) The interquartile range,
......................... years [2]
(b) The number of men who have been married for more than 26 years.
......................... [2]
Paper 4