Sets 3

4 marks · Hard

In a group of 50 students:

(a)

36 study Mathematics (set M), 31 study Biology (set B), and 20 study both subjects. Draw a Venn diagram to represent this information, labelling the numbers in each region.

[2 marks]

(b)

Calculate the probability that a randomly chosen student studies:

(i)

both Mathematics and Biology

[1 mark]

(ii)

Mathematics but not Biology

[1 mark]

Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.

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(a) 15 (M only), 11 (B only), 20 (both), 4 (neither); correctly labelled Venn diagram · must show 15, 20, 11, 4

(b(i)) 0.4 or 2/5

(b(ii)) 0.3 or 3/10

Must show: Venn diagram with correct regions (15, 11, 20, 4)

Common mistakes: Misplacing Venn diagram numbers; Ignoring the overlap; Using wrong total

Show mark scheme and worked solution

Marking points

M1 (a) · 1 mark · Finds correct values for at least 3 Venn regions

A1 (a) · 1 mark · All numbers correctly placed in the Venn diagram

B1 (b(i)) · 1 mark · Correct probability both study Mathematics and Biology (20/50)(=0.4)

B1 (b(ii)) · 1 mark · Correct probability studies Mathematics only (15/50)(=0.3)

Worked solution

Let x be only Maths; y only Biology; 20 both. x + 20 = 36, so x = 16. y + 20 = 31, so y = 11. x + y + 20 = 16 + 11 + 20 = 47. 50 - 47 = 3 (neither).

Total: Only M = 16, Only B = 11, Both = 20, Neither = 3. (Correction: Let's check) 36 - 20 = 16 (M only), 31 - 20 = 11 (B only), both = 20, so total is 16+11+20=47, rest is 3.

a) Venn diagram regions: M only = 16, B only = 11, Both = 20, Neither = 3.

b) (i) Probability both =

(ii) Probability Maths but not Biology = (Or as written above '15', so check the region numbers: 36-20=16).