A group of students at Dune Canyon High School were surveyed. They were asked which of the following products: books (B), music (M) or films (F), they downloaded from the internet.

The following results were obtained:

100 students downloaded music;
95 students downloaded films;
68 students downloaded films and music;
52 students downloaded books and music;
50 students downloaded films and books;
40 students downloaded all three products;
8 students downloaded books only;
25 students downloaded none of the three products.

Use the above information to complete a Venn diagram.

Calculate the number of students who were surveyed.

i)     On your Venn diagram, shade the set \({\left( {F \cup M} \right) \cap B'}\) . Do not shade any labels or values on the diagram.

ii)    Find \(n\left( {\left( {F \cup M} \right) \cap B'} \right)\) .

A student who was surveyed is chosen at random.

Find the probability that

(i)     the student downloaded music;

(ii)    the student downloaded books, given that they had not downloaded films;

(iii)   the student downloaded at least two of the products.

Dune Canyon High School has 850 students.

Find the expected number of students at Dune Canyon High School that downloaded music.

(A1)(A1)(A1)(A1)(A1)

Note: Award (A1) for labelled sets B, M and F inside a universal set (label U is not required).
Award (A1) for \({\text{40}}\) in central area.
Award (A1) for correct \({\text{10, 12, 28}}\) in the other intersecting regions.
Award (A1) for \({\text{8, 20}}\) and \({\text{17}}\) in correct regions.
Award (A1) for correct \({\text{25}}\).

\(8 + 12 + 20 + 10 + 40 + 28 + 17 + 25\)       (M1)

\( = 160\)        (A1)(ft)(G2)

Note: Award (M1) for adding all values. Follow through from their Venn Diagram.

i)

(A1)

ii)     \(20 + 28 + 17\)       (M1)

OR

\((100 + 95 - 68) - (10 + 40 + 12)\)       (M1)

\( = 65\)       (A1)(ft)(G2)

Note: Award (M1) for addition of the correct values from their diagram. Follow through from part (a) or (b) and part (c)(i).

i)     \(\frac{{100}}{{160}}\,\,\left( {\frac{5}{8},\,\,0.625,\,\,62.5\% } \right)\)        (A1)(A1)(ft)

Note: Award (A1) for correct numerator, (A1)(ft) for correct denominator. Follow through from part (b).

ii)    \(\frac{{20}}{{65}}\,\,\left( {\frac{4}{{13}},\,\,0.308,\,\,30.8\% } \right)\,\,\,(0.307692...)\)       (A1)(ft)(A1)(ft)

Note: Award (A1)(ft) for correct numerator, (A1)(ft) for correct denominator. Follow through from part (a).

iii)   \(\frac{{90}}{{160}}\,\,\left( {\frac{9}{{16}},\,\,0.563,\,\,56.3\% } \right)\,\,\,(0.5625)\)       (A1)(ft)(A1)(ft)

Note: Award (A1)(ft) for correct numerator, (A1)(ft) for correct denominator. Follow through from parts (a) and (b).

\(\frac{{100}}{{160}} \times 850\)       (M1)

Note: Award (M1) for their part (d)(i) multiplied by \(850\).

\( = 531\,\,\,(531.25)\)       (A1)(ft)(G2)

Note: Follow through from part (d)(i) or from part (b).

Question 2: Venn diagram, probability and expected value.
Candidates were able to draw a labelled Venn diagram and correctly place 40 and 25. A common mistake was to misinterpret the intersection of sets. In most cases this resulted only in the loss of 2 marks. Many added correctly the values in their diagram and follow-through marks were awarded irrespective of working seen, allowing the candidates who produced an incorrect diagram to obtain full marks further in the question. The most common error was not including the 25 in their total. For part (c) many correct areas, but also many incorrect areas were seen. Again follow-through marks were awarded for part (c)(ii) irrespective of working shown. Some candidates just counted the number of regions. The simple probabilities in (d)(i) and (iii) were answered correctly by the majority, the conditional probability in part (d)(ii) had very often an incorrect denominator. Some candidates with an incorrect Venn diagram lost a mark in part (d)(i) as they used the value from their diagram for the numerator and not the 100 given in the question. Candidates should be aware that when values are given in the question those should always be used and follow-through marks will not be available. Many were able to find the expected number of students in part (e). Some candidates lost follow-through marks for not showing their working here.

Question 2: Venn diagram, probability and expected value.

Candidates were able to draw a labelled Venn diagram and correctly place 40 and 25. A common mistake was to misinterpret the intersection of sets. In most cases this resulted only in the loss of 2 marks. Many added correctly the values in their diagram and follow-through marks were awarded irrespective of working seen, allowing the candidates who produced an incorrect diagram to obtain full marks further in the question. The most common error was not including the 25 in their total. For part (c) many correct areas, but also many incorrect areas were seen. Again follow-through marks were awarded for part (c)(ii) irrespective of working shown. Some candidates just counted the number of regions. The simple probabilities in (d)(i) and (iii) were answered correctly by the majority, the conditional probability in part (d)(ii) had very often an incorrect denominator. Some candidates with an incorrect Venn diagram lost a mark in part (d)(i) as they used the value from their diagram for the numerator and not the 100 given in the question. Candidates should be aware that when values are given in the question those should always be used and follow-through marks will not be available. Many were able to find the expected number of students in part (e). Some candidates lost follow-through marks for not showing their working here.

Question 2: Venn diagram, probability and expected value.

Candidates were able to draw a labelled Venn diagram and correctly place 40 and 25. A common mistake was to misinterpret the intersection of sets. In most cases this resulted only in the loss of 2 marks. Many added correctly the values in their diagram and follow-through marks were awarded irrespective of working seen, allowing the candidates who produced an incorrect diagram to obtain full marks further in the question. The most common error was not including the 25 in their total. For part (c) many correct areas, but also many incorrect areas were seen. Again follow-through marks were awarded for part (c)(ii) irrespective of working shown. Some candidates just counted the number of regions. The simple probabilities in (d)(i) and (iii) were answered correctly by the majority, the conditional probability in part (d)(ii) had very often an incorrect denominator. Some candidates with an incorrect Venn diagram lost a mark in part (d)(i) as they used the value from their diagram for the numerator and not the 100 given in the question. Candidates should be aware that when values are given in the question those should always be used and follow-through marks will not be available. Many were able to find the expected number of students in part (e). Some candidates lost follow-through marks for not showing their working here.

Question 2: Venn diagram, probability and expected value.

Candidates were able to draw a labelled Venn diagram and correctly place 40 and 25. A common mistake was to misinterpret the intersection of sets. In most cases this resulted only in the loss of 2 marks. Many added correctly the values in their diagram and follow-through marks were awarded irrespective of working seen, allowing the candidates who produced an incorrect diagram to obtain full marks further in the question. The most common error was not including the 25 in their total. For part (c) many correct areas, but also many incorrect areas were seen. Again follow-through marks were awarded for part (c)(ii) irrespective of working shown. Some candidates just counted the number of regions. The simple probabilities in (d)(i) and (iii) were answered correctly by the majority, the conditional probability in part (d)(ii) had very often an incorrect denominator. Some candidates with an incorrect Venn diagram lost a mark in part (d)(i) as they used the value from their diagram for the numerator and not the 100 given in the question. Candidates should be aware that when values are given in the question those should always be used and follow-through marks will not be available. Many were able to find the expected number of students in part (e). Some candidates lost follow-through marks for not showing their working here.

Question 2: Venn diagram, probability and expected value.

Candidates were able to draw a labelled Venn diagram and correctly place 40 and 25. A common mistake was to misinterpret the intersection of sets. In most cases this resulted only in the loss of 2 marks. Many added correctly the values in their diagram and follow-through marks were awarded irrespective of working seen, allowing the candidates who produced an incorrect diagram to obtain full marks further in the question. The most common error was not including the 25 in their total. For part (c) many correct areas, but also many incorrect areas were seen. Again follow-through marks were awarded for part (c)(ii) irrespective of working shown. Some candidates just counted the number of regions. The simple probabilities in (d)(i) and (iii) were answered correctly by the majority, the conditional probability in part (d)(ii) had very often an incorrect denominator. Some candidates with an incorrect Venn diagram lost a mark in part (d)(i) as they used the value from their diagram for the numerator and not the 100 given in the question. Candidates should be aware that when values are given in the question those should always be used and follow-through marks will not be available. Many were able to find the expected number of students in part (e). Some candidates lost follow-through marks for not showing their working here.