Unit 10

10 marks · Normal

This question involves Venn diagrams and tree diagrams.

(a)

Sets , and are defined as follows:

List all the elements of sets , , , and .

[2 marks]

(b)

Represent the sets , and in a Venn diagram. Clearly indicate and label the region .

[2 marks]

(c)

A number is selected at random from set .

Draw a tree diagram to show the possible outcomes and probabilities for 'multiple of 2' and 'multiple of 3' in set .

[2 marks]

(d)

Calculate the probability that a number chosen at random from set is both a multiple of 2 and a multiple of 3.

[2 marks]

(e)

Using your tree diagram or otherwise, calculate the probability that the number chosen is a multiple of 2 or a multiple of 3 (or both). Give your answer as a fraction in its simplest form.

[2 marks]

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.

Show answer

(a) A = {1,2,3,4,5,6,7,8,9}; B ∩ A = {2,4,6,8}; C ∩ A = {3,6,9}; B ∩ C ∩ A = {6} · must show All set intersections listed

(b) Venn diagram with circles labeled A, B, C; B ∩ C ∩ A region contains only 6 · must show Venn diagram with sets A, B, C correctly shown; overlap with correct numbers

(c) Tree diagram with first branching for 'multiple of 2' (prob 4/9), 'not multiple of 2' (5/9), then from each, branching for 'multiple of 3' (3/9 or 2/5, as per correct computation); all paths labelled · must show Tree diagram structure as in solution

(d) Probability = 1/9 (only number is 6)

(e) Probability = number in (B ∪ C)/9 = 6/9 = 2/3

Must show: Venn diagram with regions clearly labelled, Tree diagram showing correct structure

Common mistakes: Incorrect overlap in Venn diagram; Not simplifying probability fractions

Show mark scheme and worked solution

Marking points

B1 (a) · 1 mark · All elements of A and intersections listed correctly.

B1 (a) · 1 mark · Elements of B ∩ A, C ∩ A, B ∩ C ∩ A correct.

D1 (b) · 1 mark · Correct Venn diagram structure with sets labelled.

D1 (b) · 1 mark · Correct regions identified, with 6 in B ∩ C ∩ A region.

D1 (c) · 1 mark · Tree diagram shows probabilities for 'multiple of 2' and 'multiple of 3' accurately.

D1 (c) · 1 mark · Correct labels on branches.

M1 (d) · 1 mark · Correctly determines P(multiple of 2 and multiple of 3). (follow-through)

A1 (d) · 1 mark · Gives answer as 1/9. (follow-through)

M1 (e) · 1 mark · Finds P(multiple of 2 or multiple of 3) using correct method (sum, or set formula). (follow-through)

A1 (e) · 1 mark · 2/3 (simplest form). (follow-through)

Worked solution

(a) ,

B (even numbers) within A: {2,4,6,8}; C (multiples of 3) within A: {3,6,9}; B ∩ C ∩ A: {6}

(b) Venn diagram: Circles A, B, C. Place 2,4,8 in B only; 3,9 in C only; 6 where all three overlap; rest outside or in only A.

(c) Tree diagram: First branch: Is number a multiple of 2? Yes: numbers 2,4,6,8 (4/9); No: 1,3,5,7,9 (5/9). Second: each branch, further split: Is number a multiple of 3? Only 6 is both: probability 1/9 etc.

(d) Probability = 1/9 (only the number 6 is in all sets).

(e) Numbers that are multiples of 2 or 3: 2,3,4,6,8,9 = 6 numbers. Probability 6/9 = 2/3.