IGCSE Maths · Edexcel 4XMA1 · Unit 1 Higher · Statistics

Two-Way Tables — Worked Solutions

Full worked solutions to all 11 questions (34 marks), set out the way an Edexcel examiner wants to see them.

Section A — Warm-up (grades 4–5)

(3 marks)35 students chose a sport. Complete the two-way table.
Boys Girls Total
Tennis 8 20
Football 6
Total 17 35
Solution
1. Tennis row: girls = 20 − 8 = 12.
2. Boys column: football = 17 − 8 = 9.
3. Girls total = 35 − 17 = 18, and football girls = 6 ✓ (12 + 6 = 18).
4. Football total = 9 + 6 = 15.
Boys Girls Total
Tennis 8 12 20
Football 9 6 15
Total 17 18 35
Tennis girls 12, football boys 9, football total 15, girls total 18
(2 marks)Use your completed table from Question 1. A student is chosen at random.
(a) Find the probability that the student is a girl who plays tennis.
(b) Find the probability that the student plays football.
Solution
1. (a) 12 girls play tennis out of 35: P = 1235.
2. (b) 15 play football: P = 1535 = 37.
(a) 12/35 (b) 3/7
(3 marks)Complete the table showing how 80 students travel to school.
Walk Bus Car Total
Year 10 12 9 40
Year 11 15
Total 25 80
Solution
1. Year 10 bus = 40 − 12 − 9 = 19.
2. Year 11 total = 80 − 40 = 40; Year 11 walk = 25 − 12 = 13.
3. Year 11 car = 40 − 13 − 15 = 12.
4. Bus total = 34; car total = 21.
Y10 bus 19; Y11: 13, 15, 12, total 40; totals 25, 34, 21

	Boys	Girls	Total
Tennis	8		20
Football		6
Total	17		35

	Boys	Girls	Total
Tennis	8	12	20
Football	9	6	15
Total	17	18	35

	Walk	Bus	Car	Total
Year 10	12		9	40
Year 11		15
Total	25			80

Section B — Core (grades 5–7)

(4 marks)120 students each chose one language: French, Spanish or Mandarin.
64 of the students are girls. 23 girls chose French. 31 boys chose Spanish. 40 students chose Mandarin, and 18 of these are boys.
Draw a two-way table and use it to find how many girls chose Spanish.
Solution
1. Boys = 120 − 64 = 56.
2. Boys French = 56 − 31 − 18 = 7.
3. Girls Mandarin = 40 − 18 = 22.
4. Girls Spanish = 64 − 23 − 22 = 19.
19
(2 marks)Using Question 4, one of the students who chose Mandarin is picked at random. Find the probability that this student is a girl.
Solution
1. Choose only from the 40 Mandarin students. 22 of them are girls.
2. P = 2240 = 1120.
11/20
(3 marks)100 people were asked how they prefer to watch films.
Age (years) Cinema Streaming Neither Total
0–17 14 9 26
18–39 18 6 35
40+ 5 22
Total 39 100
(a) Complete the table.
(b) What fraction of the people aged 40 or over prefer streaming?
Solution
1. 0–17 neither = 26 − 14 − 9 = 3; 18–39 cinema = 35 − 18 − 6 = 11.
2. 40+ total = 100 − 26 − 35 = 39; 40+ streaming = 39 − 5 − 22 = 12.
3. Check streaming total: 9 + 18 + 12 = 39 ✓. Cinema total = 30; neither total = 31.
4. (b) 1239 = 413.
(b) 4/13
(4 marks)250 members of a sports club were surveyed. 40% of the members are juniors. 60% of the juniors swim. 75 of the adults swim.
(a) Complete a two-way table for this information.
(b) A member is chosen at random. Find the probability that they swim.
Solution
1. Juniors = 40% of 250 = 100; adults = 150.
2. Juniors who swim = 60% of 100 = 60, so 40 juniors don’t swim.
3. Adults: 75 swim, so 150 − 75 = 75 don’t swim.
Swim Don’t swim Total
Juniors 60 40 100
Adults 75 75 150
Total 135 115 250
1. (b) P(swims) = 135250 = 2750.
(b) 27/50
(2 marks)The club in Question 7 opens a new branch with 400 members. Assuming the same pattern, estimate how many of them swim.
Solution
1. Expected frequency = probability × number of trials.
2. 400 × 2750 = 216.
216 members

Age (years)	Cinema	Streaming	Neither	Total
0–17	14	9		26
18–39		18	6	35
40+	5		22
Total		39		100

	Swim	Don’t swim	Total
Juniors	60	40	100
Adults	75	75	150
Total	135	115	250

Section C — Stretch (grades 8–9)

(4 marks)80 people chose Plan A or Plan B, and each person is either a new or an existing customer.
Plan A Plan B Total
New x 2x
Existing x + 5 15
Total 80
(a) Find x.
(b) Find the probability that a person chosen at random from the Plan A customers is a new customer.
Solution
1. All four cells add to 80: x + 2x + (x + 5) + 15 = 80.
2. 4x + 20 = 80, so x = 15.
3. Plan A: new = 15, existing = 20, total 35.
4. (b) P = 1535 = 37.
(a) x = 15 (b) 3/7
(4 marks)A chess club has 96 members. The ratio of boys to girls is 3 : 5. One quarter of the boys and 3/10 of the girls play in tournaments.
A tournament player is chosen at random. Find the probability that this player is a girl.
Solution
1. Boys = 38 × 96 = 36; girls = 58 × 96 = 60.
2. Boys in tournaments = 14 × 36 = 9; girls = 310 × 60 = 18.
3. Tournament players = 9 + 18 = 27.
4. P(girl | tournament player) = 1827 = 23.
2/3
(3 marks)Draw a Venn diagram that shows the same information as this table, using C = {has a cat} and D = {has a dog}.
Cat No cat Total
Dog 6 10 16
No dog 14 10 24
Total 20 20 40
Solution
1. C ∩ D = cat and dog = 6.
2. C only = cat, no dog = 14. D only = dog, no cat = 10.
3. Outside = neither = 10. Check: 14 + 6 + 10 + 10 = 40 ✓
C only 14, both 6, D only 10, outside 10

	Plan A	Plan B	Total
New	x	2x
Existing	x + 5	15
Total			80

	Cat	No cat	Total
Dog	6	10	16
No dog	14	10	24
Total	20	20	40

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