Unit 7
5 marks · NormalWrite the decimal 0.1\dot{7} as a fraction in its simplest form.
A bag contains 12 discs, each labelled with a different integer from 1 to 12. A disc is selected at random.
Let D be the event ‘the disc number is a divisor of the denominator of your answer to part (a)’. Let E be the event ‘the disc number is even’.
Represent this information using a Venn diagram.
Draw the Venn diagram with the universal set consisting of the numbers 1 to 12. Clearly label each set and shade the region representing D \cap E.
Find the probability that a randomly chosen disc shows a number that is both a divisor of the denominator of your answer to part (a) and even.
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) \frac{17}{99}
(b) Venn diagram: Universal set \{1,2,3,...,12\}, Set D (divisors of 99: 1,3,9,11,33,99; restrict to 1,3,9,11 since only 1-12), Set E (even numbers in 1-12: 2,4,6,8,10,12). D \cap E = \emptyset. · must show Universal set 1-12, Set D (1,3,9,11), Set E (2,4,6,8,10,12), Indicate D \cap E is empty or shaded accordingly
(c) 0
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Marking points
M1 (a) · 1 mark · Sets x = 0.171717...; multiplies by 100, subtracts x; gets a correct equation.
A1 (a) · 1 mark · Correctly simplifies and gives the answer as in lowest terms.
B1 (b) · 1 mark · Correctly draws or describes a Venn diagram for sets D and E, showing elements in each, from 1 to 12, and labels the sets.
B2 (b) · 1 mark · Clearly indicates/shades the region D \cap E (intersection), showing that D \cap E = \emptyset or shading nothing in the intersection.
B3 (c) · 1 mark · States the probability as 0, using their Venn diagram. (follow-through)
Worked solution
(a) Let . Since two digits repeat, multiply both sides by 100:
Subtract the original equation from this:
(b) The denominator of is 99. The divisors of 99 from 1 to 12 are 1, 3, 9, 11.
The even numbers from 1 to 12 are 2, 4, 6, 8, 10, and 12.
No number in 1–12 is both a divisor of 99 and even, so D \cap E is empty.
(c) The probability of a number being both even and a divisor of 99 (from 1 to 12) is 0, since the intersection is empty.