Probability and Data Handling Worksheet with Answers

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13+ · Mathematics

CE.M12 Probability and Data Handling

ISEB CE ISEB CE 13+ Mathematics · Calculator allowed · about 70 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

1

A bag contains 7 red counters and 4 blue counters, and nothing else. Priya takes one counter from the bag at random. Write down the probability that she takes a blue counter.

(Total for Question 1 is 1 mark)

2

A fair spinner has 8 equal sections numbered 1 to 8. Ravi spins the spinner once. Calculate the probability that the spinner lands on a multiple of 3.

(Total for Question 2 is 2 marks)

3

The ages, in years, of 9 dogs at a rescue centre are recorded below:
1, 2, 2, 3, 3, 3, 5, 6, 8

(a)Write down the mode of the ages.(1)

(b)Calculate the range of the ages.(1)

(Total for Question 3 is 2 marks)

4

The table shows the number of pets owned by 20 pupils in a class.
Number of pets: 0, 1, 2, 3
Frequency: 6, 9, 3, 2

(a)Write down the modal number of pets.(1)

(b)Find the number of pupils who own at least 2 pets.(1)

(c)Calculate the range of the number of pets owned.(1)

(Total for Question 4 is 3 marks)

5

In her last 5 matches, a tennis player recorded the following number of aces: 2, 3, 1, 4, 5

(a)Calculate the mean number of aces per match.(2)

(b)In her sixth match she records 3 aces. Without recalculating the whole mean, state whether the mean will increase, decrease or stay the same, giving a reason.(1)

(Total for Question 5 is 3 marks)

6

A six-sided die is suspected of being biased. It is rolled 80 times, and it lands on a 6 on 28 of those rolls.

(a)Find the relative frequency of rolling a 6.(2)

(b)For a fair die, the probability of rolling a 6 is 1/6. Comment on whether these results suggest the die is biased in favour of rolling a 6.(1)

(Total for Question 6 is 3 marks)

7

60 pupils in Year 8 were asked whether they walk or take the bus to school, and whether they are members of the football club. Some of the results are shown in the table below (numbers of pupils).

Football club Not football club Total
Walk 12 ? 34
Bus ? 10 ?
Total 28 ? 60

(a)Complete the table.(2)

(b)A pupil is chosen at random from the 60 pupils. Find the probability that the pupil takes the bus and is not in the football club, giving your answer as a fraction in its simplest form.(2)

(Total for Question 7 is 4 marks)

8

In a survey, 90 pupils named their favourite school subject. The results were:
Maths: 25 pupils
Science: 20 pupils
English: 15 pupils
Art: 10 pupils
Other: 20 pupils
A pie chart is to be drawn to show this information.

(a)Calculate the angle of the pie-chart sector that represents pupils who chose Maths.(2)

(b)Calculate the angle of the pie-chart sector that represents pupils who chose 'Other'.(2)

(Total for Question 8 is 4 marks)

9

The table shows the number of texts sent in one day by 30 pupils.
Number of texts (x): 0, 1, 2, 3, 4
Frequency (f): 3, 8, 10, 6, 3
Calculate the mean number of texts sent per pupil, giving your answer correct to 3 significant figures.

(Total for Question 9 is 4 marks)

10

The probability that a particular machine produces a faulty component on any given day is 0.35.

(a)Write down the probability that the machine does not produce a faulty component on a given day.(1)

(b)Assuming the outcomes on different days are independent, find the probability that the machine produces a faulty component on both of the next two days.(2)

(Total for Question 10 is 3 marks)

11

The stem-and-leaf diagram below shows the times, in seconds, taken by 15 pupils to complete a puzzle.

Stem	Leaf
1	2 5 8 9
2	0 1 3 6
3	2 5 7
4	0 1 2 3

Key: 1 | 2 means 12 seconds

(a)Write down the number of pupils who took longer than 30 seconds.(1)

(b)Find the median time taken.(2)

(c)Calculate the range of the times.(1)

(Total for Question 11 is 4 marks)

12

A jar contains 3 lemon sweets and 9 orange sweets, and nothing else (12 sweets in total). Aisha takes a sweet at random and does not replace it. She then takes a second sweet at random.

(a)Write down the probability that the first sweet is a lemon sweet.(1)

(b)Find the probability that both sweets taken are lemon sweets.(2)

(c)Find the probability that the two sweets taken are different flavours.(2)

(Total for Question 12 is 5 marks)

13

A pictogram shows the number of books read in a month by four pupils. Each full book symbol represents 4 books.
Sam: 3 symbols
Priya: 2 symbols and a half symbol
Tom: 4 symbols
Layla: 1 symbol and a half symbol

(a)Write down the number of books read by Layla.(1)

(b)Find the total number of books read by all four pupils, and use this to calculate the mean number of books read per pupil.(2)

(Total for Question 13 is 3 marks)

14

The table shows the number of hours of sunshine recorded at a seaside kiosk and the number of ice creams sold, for 8 different days.
Hours of sunshine: 1, 2, 3, 4, 5, 6, 7, 8
Ice creams sold: 38, 45, 52, 60, 64, 72, 80, 88

(a)State the type of correlation shown by this data.(1)

(b)Interpret what this correlation means in the context of the data.(1)

(c)On one day there were 4.5 hours of sunshine. Use the pattern in the table to estimate the number of ice creams sold on that day.(2)

(Total for Question 14 is 4 marks)

15

In a survey of 40 pupils, 22 attend the Music club, 15 attend the Drama club, and 9 attend both clubs.

(a)Find the number of pupils who attend the Music club only.(1)

(b)Find the number of pupils who attend neither club.(1)

(c)A pupil is chosen at random from the 40 pupils. Find the probability that this pupil attends the Drama club only, giving your answer as a fraction in its simplest form.(1)

(Total for Question 15 is 3 marks)

16

The table shows, for three school forms, the number of pupils who own a pet.
Form 8A: 18 out of 25 pupils own a pet.
Form 8B: 15 out of 20 pupils own a pet.
Form 8C: 21 out of 30 pupils own a pet.

(a)Determine which form has the greatest proportion of pupils who own a pet. Show your working.(2)

(b)A pupil is chosen at random from Form 8C. Find the probability, as a decimal, that this pupil does not own a pet.(2)

(Total for Question 16 is 4 marks)

17

Two pupils, Ffion and Rosa, each sat 5 weekly spelling tests, each marked out of 10. Their scores were:
Ffion: 6, 7, 5, 8, 9
Rosa: 7, 7, 6, 8, 7

(a)Calculate the mean score for each pupil.(2)

(b)Calculate the range of each pupil's scores, and use your answers to compare the consistency of the two pupils.(2)

(Total for Question 17 is 4 marks)

18

A biased coin is designed so that the probability of it landing on heads is 0.6. The coin is tossed 3 times.

(a)Find the probability that all three tosses land on heads.(2)

(b)Find the probability of getting at least one tails in the three tosses.(3)

(Total for Question 18 is 5 marks)

19

The table shows the masses, in kg, of 40 parcels processed at a delivery depot.
Mass (m kg): 5 ≤ m < 10, frequency 6
10 ≤ m < 15, frequency 14
15 ≤ m < 20, frequency 12
20 ≤ m < 25, frequency 8

(a)Write down the modal class.(1)

(b)Calculate an estimate for the mean mass of a parcel.(3)

(c)Explain why your answer to part (b) is only an estimate.(1)

(Total for Question 19 is 5 marks)

20

A drawer contains 5 black socks and 7 white socks, and nothing else (12 socks in total). Two socks are selected at random, one after the other, without replacement.

(a)Find the probability that both socks selected are black.(2)

(b)Find the probability that exactly one of the two socks selected is black.(2)

(c)If this experiment (selecting 2 socks without replacement) were repeated 66 times, use your answer to part (b) to find how many times you would expect exactly one black sock to be selected.(2)

(Total for Question 20 is 6 marks)

Mark scheme · CE.M12 Probability and Data Handling

Question 1

B1 4/11 oe (accept 0.36 awrt, decimals or percentages equivalent)
Answer: 4/11

Question 2

M1 identifies the multiples of 3 from 1 to 8 as 3 and 6 (2 outcomes)
A1 1/4 oe (accept 2/8, 0.25, 25%)
Answer: 1/4

Question 3

(a) B1 3 cao
(a) Answer: 3
(b) B1 7 (8 - 1) cao
(b) Answer: 7

Question 4

(a) B1 1 cao
(a) Answer: 1
(b) B1 5 (3 + 2) cao
(b) Answer: 5
(c) B1 3 (3 - 0) cao
(c) Answer: 3

Question 5

(a) M1 sum = 2 + 3 + 1 + 4 + 5 = 15
(a) A1 mean = 15 / 5 = 3
(a) Answer: 3
(b) B1 stays the same, because 3 equals the current mean of 3, oe
(b) Answer: Stays the same

Question 6

(a) M1 sets up 28/80
(a) A1 0.35 oe (7/20, 35%)
(a) Answer: 0.35 (7/20)
(b) B1 valid comment comparing 0.35 to 1/6 (approx 0.167), e.g. the observed relative frequency is roughly double the fair-die probability, so the results do suggest bias towards 6, oe
(b) Answer: Yes - 0.35 is much higher than the fair-die probability of 1/6 (approx 0.167), suggesting bias towards 6.

Question 7

(a) M1 at least two of the four missing values found correctly
(a) A1 all four missing values correct: Walk & Not football = 22; Bus & Football = 16; Bus total = 26 (so Bus & Not football = 10); Not football total = 32
(a) Answer: Walk & Not football = 22; Bus & Football = 16; Bus total = 26; Not football total = 32
(b) M1 identifies Bus & Not football = 10, so probability = 10/60
(b) A1 1/6 oe, correctly simplified
(b) Answer: 1/6

Question 8

(a) M1 sets up (25/90) x 360
(a) A1 100 degrees
(a) Answer: 100 degrees
(b) M1 sets up (20/90) x 360
(b) A1 80 degrees
(b) Answer: 80 degrees

Question 9

M1 correct fx values: 0, 8, 20, 18, 12
M1 sum of fx = 58
M1 divides by the total frequency, 30
A1 1.93 awrt (accept 29/15 oe)
Answer: 1.93 (3sf)

Question 10

(a) B1 0.65
(a) Answer: 0.65
(b) M1 0.35 x 0.35
(b) A1 0.1225 (accept 0.123 to 3sf) oe
(b) Answer: 0.1225

Question 11

(a) B1 7
(a) Answer: 7
(b) M1 identifies the median position as the 8th value (n = 15, so (15+1)/2 = 8th)
(b) A1 26 seconds
(b) Answer: 26 seconds
(c) B1 31 seconds (43 - 12)
(c) Answer: 31 seconds

Question 12

(a) B1 1/4 oe (3/12, 0.25)
(a) Answer: 1/4
(b) M1 3/12 x 2/11
(b) A1 1/22 oe (accept 6/132, awrt 0.0455)
(b) Answer: 1/22
(c) M1 finds both P(lemon then orange) = 3/12 x 9/11 = 27/132 and P(orange then lemon) = 9/12 x 3/11 = 27/132
(c) A1 9/22 oe (accept 54/132, awrt 0.409)
(c) Answer: 9/22

Question 13

(a) B1 6
(a) Answer: 6
(b) M1 total = 12 + 10 + 16 + 6 = 44
(b) A1 mean = 44 / 4 = 11
(b) Answer: 44 books total; mean = 11

Question 14

(a) B1 positive correlation
(a) Answer: Positive correlation
(b) B1 more hours of sunshine is associated with more ice creams sold, oe
(b) Answer: More hours of sunshine are associated with a higher number of ice creams sold.
(c) M1 sensible interpolation between the values for 4 hours (60) and 5 hours (64), e.g. midpoint
(c) A1 answer in the range 60-64 inclusive, e.g. 62, ft from a valid method
(c) Answer: 62 (accept any value 60-64 with valid reasoning)

Question 15

(a) B1 13
(a) Answer: 13
(b) B1 12
(b) Answer: 12
(c) B1 3/20 oe (6/40)
(c) Answer: 3/20

Question 16

(a) M1 calculates at least two of the three proportions correctly (8A = 72%, 8B = 75%, 8C = 70%)
(a) A1 correctly identifies Form 8B, with all three proportions shown
(a) Answer: Form 8B (75%)
(b) M1 9/30 (pupils without a pet = 30 - 21 = 9)
(b) A1 0.3 oe (3/10)
(b) Answer: 0.3

Question 17

(a) M1 correct totals: Ffion = 35, Rosa = 35
(a) A1 both means = 7
(a) Answer: Ffion mean = 7; Rosa mean = 7
(b) M1 both ranges correct: Ffion = 4, Rosa = 2
(b) A1 correct comparison: Rosa is more consistent (smaller range), despite the equal means
(b) Answer: Ffion range = 4; Rosa range = 2; Rosa is more consistent.

Question 18

(a) M1 0.6 x 0.6 x 0.6
(a) A1 0.216
(a) Answer: 0.216
(b) M1 recognises 'at least one tails' is the complement of 'all heads'
(b) M1 uses P(all heads) = 0.216 (ft from part a)
(b) A1 0.784
(b) Answer: 0.784

Question 19

(a) B1 10 ≤ m < 15
(a) Answer: 10 ≤ m < 15
(b) M1 correct midpoints used: 7.5, 12.5, 17.5, 22.5
(b) M1 sum of f x midpoint = 610 (ft from midpoints)
(b) A1 mean estimate = 15.25 kg (accept awrt 15.3)
(b) Answer: 15.25 kg
(c) B1 because the exact mass of each parcel within a class interval is unknown, only the interval is known, so the midpoint is used as an approximation, oe
(c) Answer: The exact individual masses are not known, only the class intervals, so midpoints (an approximation) had to be used.

Question 20

(a) M1 5/12 x 4/11
(a) A1 5/33 oe (accept 20/132)
(a) Answer: 5/33
(b) M1 both P(black then white) = 5/12 x 7/11 = 35/132 and P(white then black) = 7/12 x 5/11 = 35/132 found
(b) A1 35/66 oe (accept 70/132, awrt 0.530)
(b) Answer: 35/66
(c) M1 sets up 35/66 x 66
(c) A1 35
(c) Answer: 35

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