Statistics and Probability Worksheet with Answers

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13+ · Mathematics

CE.M6 Statistics and Probability

ISEB COMMON ENTRANCE CE.M6 · Calculator allowed · about 90 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

1

The list below shows the shoe sizes of 10 pupils in a Year 8 form group:
4, 5, 5, 6, 7, 5, 8, 6, 5, 9

(a)Write down the mode of the shoe sizes.(1)

(b)Work out the range of the shoe sizes.(1)

(Total for Question 1 is 2 marks)

2

Five friends recorded how many text messages they sent yesterday: 12, 15, 18, 9, 16.

(Total for Question 2 is 2 marks)

3

A bag contains 5 red counters, 3 blue counters and 2 green counters. A counter is taken from the bag at random.

(a)Write down the probability that the counter is red.(1)

(b)Find the probability that the counter is not blue.(1)

(Total for Question 3 is 2 marks)

4

Two fair coins are tossed together.

(a)List all the possible outcomes.(1)

(b)Find the probability of getting exactly one head.(2)

(Total for Question 4 is 3 marks)

5

A hockey team played 20 matches. The table shows the number of goals scored per match.
Goals scored: 0, 1, 2, 3
Frequency: 5, 8, 4, 3

(Total for Question 5 is 3 marks)

6

A survey asked whether Year 8 students own a bicycle. Some results are shown in the two-way table.
Yes No Total
Boys 18 12 30
Girls ? 10 25
Total ? 22 55

(a)Find the number of girls who own a bicycle.(1)

(b)A student is chosen at random from the whole group. Find the probability that the student is a girl who owns a bicycle. Give your answer as a fraction in its simplest form.(2)

(Total for Question 6 is 3 marks)

7

In a survey, 72 pupils were asked to name their favourite sport. The results are shown in a pie chart with the following angles: Football 150 degrees, Netball 90 degrees, Tennis 60 degrees, Other 60 degrees.

(a)Calculate the number of pupils who chose Tennis.(2)

(b)What fraction of the pupils chose Football or Netball? Give your answer in its simplest form.(2)

(Total for Question 7 is 4 marks)

8

A biased spinner has four colours: red, blue, green and yellow.
P(red) = 0.3, P(blue) = 0.25, P(green) = 0.2.

(a)Find P(yellow).(1)

(b)The spinner is spun 200 times. Work out an estimate for the number of times it lands on red.(2)

(Total for Question 8 is 3 marks)

9

A spinner is spun 50 times and the results are recorded: Red 18, Blue 22, Green 10.

(a)Find the relative frequency (estimated probability) of landing on green.(1)

(b)The spinner is now spun a further 300 times. Estimate the number of times it will land on green.(2)

(Total for Question 9 is 3 marks)

10

Two fair six-sided dice are rolled and the two scores are added together.

(a)Find the probability that the total is 7.(2)

(b)Find the probability that the total is at least 10.(2)

(Total for Question 10 is 4 marks)

11

In a survey of 30 pupils, 18 play football (F), 15 play hockey (H), and 8 play both football and hockey.

(a)Draw a Venn diagram to show this information, and find the number of pupils who play neither sport.(2)

(b)Find the probability that a pupil chosen at random plays exactly one of the two sports.(2)

(Total for Question 11 is 4 marks)

12

A teacher plots a scatter graph of hours spent revising (horizontal axis) against test score out of 100 (vertical axis) for 15 pupils. The points show a clear pattern rising from bottom left to top right, and a line of best fit has been drawn through the data. The line of best fit passes through the points (0, 40) and (10, 90).

(a)Describe the type of correlation shown by the scatter graph.(1)

(b)One pupil revised for 6 hours. Using the line of best fit, estimate this pupil's test score.(2)

(Total for Question 12 is 3 marks)

13

The stem-and-leaf diagram shows the test scores (out of 50) of 13 pupils in a class.

(a)Find the median test score.(1)

(b)Find the range of the test scores.(1)

(c)Find the number of pupils who scored more than 25 marks.(1)

(Total for Question 13 is 3 marks)

14

The table shows the time, t minutes, taken by 20 pupils to complete a fun run.

Time (t minutes)	Frequency
10 ≤ t < 15	4
15 ≤ t < 20	9
20 ≤ t < 25	5
25 ≤ t < 30	2

(a)Write down the midpoint of the class 15 ≤ t < 20.(1)

(b)Calculate an estimate of the mean time taken.(3)

(Total for Question 14 is 4 marks)

15

Two athletes, Priya and Fola, each complete 5 training runs. Their times, in minutes, are:
Priya: 20, 22, 19, 21, 18
Fola: 15, 25, 12, 28, 20
Both athletes have a mean time of 20 minutes.

(a)Calculate the range of each athlete's times.(2)

(b)Given that both athletes have the same mean time, use the ranges to compare the consistency of the two athletes.(2)

(Total for Question 15 is 4 marks)

16

In a class of 28 pupils, 16 study French (F), 12 study Spanish (S), and 5 study both French and Spanish.

(a)Find the number of pupils who study neither French nor Spanish.(2)

(b)A pupil is selected at random from those who study French. Find the probability that this pupil also studies Spanish.(2)

(Total for Question 16 is 4 marks)

17

A bag contains 4 red counters and 6 blue counters. Two counters are taken from the bag at random, one after the other, without replacement.

(a)Find the probability that both counters are red.(3)

(b)Find the probability that the two counters are different colours.(3)

(Total for Question 17 is 6 marks)

18

The probability that it rains on any given Saturday in a particular town is 0.4. Whether it rains on Saturday is independent of whether it rains on Sunday, and the probability that it rains on Sunday is also 0.4.

(a)Find the probability that it rains on both Saturday and Sunday.(2)

(b)Find the probability that it rains on at least one of the two days.(3)

(Total for Question 18 is 5 marks)

19

Spinner A has three equal sections numbered 1, 2, 3. Spinner B has two equal sections numbered 1, 2. Both spinners are spun once and the two numbers are added together to give a total score.

(a)List all 6 possible total scores.(2)

(b)Find the probability that the total score is 4.(2)

(c)Find the probability that the total score is an even number.(2)

(Total for Question 19 is 6 marks)

20

A bag contains x red counters and (x + 3) blue counters. A counter is taken from the bag at random. The probability that the counter is red is 2/5.

(a)Show that 5x = 4x + 6.(2)

(b)Solve the equation to find the value of x.(1)

(c)Find the probability that the counter is blue.(2)

(Total for Question 20 is 5 marks)

Mark scheme · CE.M6 Statistics and Probability

Question 1

(a) B1 mode = 5 cao
(a) Answer: 5
(b) B1 range = 9 - 4 = 5 cao
(b) Answer: 5

Question 2

M1 sum of values: 12 + 15 + 18 + 9 + 16 = 70
A1 mean = 70 / 5 = 14 cao
Answer: 14 text messages

Question 3

(a) B1 P(red) = 5/10 = 1/2 oe
(a) Answer: 1/2
(b) B1 P(not blue) = 7/10 oe, e.g. 1 - 3/10
(b) Answer: 7/10

Question 4

(a) B1 HH, HT, TH, TT all listed
(a) Answer: HH, HT, TH, TT
(b) M1 identify 2 favourable outcomes (HT, TH) out of 4
(b) A1 P(exactly one head) = 2/4 = 1/2 oe
(b) Answer: 1/2

Question 5

M1 correct products: (0x5) + (1x8) + (2x4) + (3x3)
M1 total goals = 25 (from 0 + 8 + 8 + 9), used with total matches = 20
A1 mean = 25/20 = 1.25 goals per match cao
Answer: 1.25 goals per match

Question 6

(a) B1 15 (from 25 - 10)
(a) Answer: 15
(b) M1 15/55 seen ft from part (a)
(b) A1 3/11 cao (simplified)
(b) Answer: 3/11

Question 7

(a) M1 60/360 x 72
(a) A1 12 cao
(a) Answer: 12 pupils
(b) M1 (150 + 90)/360 seen
(b) A1 2/3 cao (simplified)
(b) Answer: 2/3

Question 8

(a) B1 0.25 cao
(a) Answer: 0.25
(b) M1 0.3 x 200
(b) A1 60 cao
(b) Answer: 60 times

Question 9

(a) B1 10/50 = 1/5 or 0.2 oe
(a) Answer: 1/5 (0.2)
(b) M1 300 x 1/5 (or 300 x 0.2) ft from part (a)
(b) A1 60 cao
(b) Answer: 60 times

Question 10

(a) M1 identify 6 favourable outcomes out of 36, e.g. (1,6)(2,5)(3,4)(4,3)(5,2)(6,1)
(a) A1 1/6 cao
(a) Answer: 1/6
(b) M1 identify 6 favourable outcomes out of 36, e.g. totals of 10, 11 and 12
(b) A1 1/6 cao
(b) Answer: 1/6

Question 11

(a) M1 correct regions found: football only = 10, hockey only = 7, both = 8
(a) A1 neither = 5 cao
(a) Answer: 5 pupils
(b) M1 (10 + 7)/30 ft from part (a)
(b) A1 17/30 cao (already in simplest form)
(b) Answer: 17/30

Question 12

(a) B1 positive correlation oe
(a) Answer: Positive correlation
(b) M1 correct gradient found: (90 - 40)/(10 - 0) = 5 marks per hour, or equivalent method
(b) A1 70 cao (accept 65 to 75 if read from a drawn graph)
(b) Answer: 70

Question 13

(a) B1 27 cao
(a) Answer: 27
(b) B1 29 cao (41 - 12)
(b) Answer: 29
(c) B1 7 cao
(c) Answer: 7 pupils

Question 14

(a) B1 17.5 cao
(a) Answer: 17.5 minutes
(b) M1 all four midpoints used: 12.5, 17.5, 22.5, 27.5
(b) M1 sum of (midpoint x frequency) = 50 + 157.5 + 112.5 + 55 = 375
(b) A1 mean = 375/20 = 18.75 minutes cao
(b) Answer: 18.75 minutes

Question 15

(a) M1 one correct range found (Priya = 4 or Fola = 16)
(a) A1 both ranges correct: Priya = 4, Fola = 16
(a) Answer: Priya: 4 minutes, Fola: 16 minutes
(b) B1 identifies Priya as more consistent, ft from part (a)
(b) B1 valid reason given: smaller range means times are less spread out / closer together (4 < 16) oe
(b) Answer: Priya is more consistent, because her range (4 minutes) is smaller than Fola's range (16 minutes), so her times are less spread out.

Question 16

(a) M1 correct method, e.g. 28 - (16 + 12 - 5)
(a) A1 5 cao
(a) Answer: 5 pupils
(b) M1 5/16 seen, using 16 as the denominator (those studying French)
(b) A1 5/16 cao (already in simplest form)
(b) Answer: 5/16

Question 17

(a) M1 P(1st red) = 4/10
(a) M1 P(2nd red given 1st red) = 3/9, and 4/10 x 3/9 used
(a) A1 2/15 cao
(a) Answer: 2/15
(b) M1 P(red then blue) = 4/10 x 6/9, or P(blue then red) = 6/10 x 4/9
(b) M1 both branches added: (4/10 x 6/9) + (6/10 x 4/9)
(b) A1 8/15 cao
(b) Answer: 8/15

Question 18

(a) M1 0.4 x 0.4
(a) A1 0.16 cao
(a) Answer: 0.16
(b) M1 P(no rain either day) = 0.6 x 0.6
(b) M1 0.6 x 0.6 = 0.36, then 1 - 0.36 used
(b) A1 0.64 cao
(b) Answer: 0.64

Question 19

(a) M1 at least 4 of the 6 outcomes correctly paired and totalled
(a) A1 all 6 totals correct: 2, 3, 3, 4, 4, 5
(a) Answer: 2, 3, 3, 4, 4, 5
(b) M1 2 favourable outcomes out of 6 identified, ft from part (a)
(b) A1 1/3 cao
(b) Answer: 1/3
(c) M1 3 favourable outcomes out of 6 identified, ft from part (a)
(c) A1 1/2 cao
(c) Answer: 1/2

Question 20

(a) M1 correct equation formed: x/(2x+3) = 2/5, then cross-multiplied to 5x = 2(2x+3)
(a) A1 cso: 2(2x+3) correctly expanded to give 5x = 4x + 6, answer printed
(a) Answer: 5x = 4x + 6 (shown)
(b) B1 x = 6 cao, ft from part (a)
(b) Answer: x = 6
(c) M1 total counters = 2(6)+3 = 15 and blue counters = 6+3 = 9 found, ft from part (b)
(c) A1 3/5 cao (simplified from 9/15)
(c) Answer: 3/5

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