| Date | May 2008 | Marks available | 2 | Reference code | 08M.2.sl.TZ1.2 |
| Level | SL only | Paper | 2 | Time zone | TZ1 |
| Command term | Calculate | Question number | 2 | Adapted from | N/A |
Question
A group of 50 students completed a questionnaire for a Mathematical Studies project. The following data was collected.
\(18\) students own a digital camera (D)
\(15\) students own an iPod (I)
\(26\) students own a cell phone (C)
\(1\) student owns all three items
\(5\) students own a digital camera and an iPod but not a cell phone
\(2\) students own a cell phone and an iPod but not a digital camera
\(3\) students own a cell phone and a digital camera but not an iPod
Claire and Kate both wish to go to the cinema but one of them has to stay at home to baby-sit.
The probability that Kate goes to the cinema is \(0.2\). If Kate does not go Claire goes.
If Kate goes to the cinema the probability that she is late home is \(0.3\).
If Claire goes to the cinema the probability that she is late home is \(0.6\).
Represent this information on a Venn diagram.
Calculate the number of students who own none of the items mentioned above.
If a student is chosen at random, write down the probability that the student owns a digital camera only.
If two students are chosen at random, calculate the probability that they both own a cell phone only.
If a student owns an iPod, write down the probability that the student also owns a digital camera.
Copy and complete the probability tree diagram below.
Calculate the probability that
(i) Kate goes to the cinema and is not late;
(ii) the person who goes to the cinema arrives home late.
Markscheme
(A1)(A1)(A1)(A1)(ft)
Note: (A1) for rectangle with 3 intersecting circles, (A1) for \(1\), (A1) for \(5\), \(3\), \(2\), (A1)(ft) for \(9\), \(7\), \(20\) if subtraction is carried out, or \(18\), \(15\), \(26\) seen by the letters D, I and C.
[4 marks]
\(50 - 47\) (M1)
Note: (M1) for subtracting their value from \(50\).
\( = 3\) (A1)(ft)(G2)
[2 marks]
\(\frac{9}{{50}}\) (A1)(ft)
[1 mark]
\(\frac{{20}}{{50}} \times \frac{{19}}{{49}}\) (A1)(ft)(M1)
\( = \frac{{38}}{{245}}{\text{ }}\left( {\frac{{380}}{{2450}}{\text{, }}0.155{\text{, }}15.5\% } \right)\) (A1)(ft)(G2)
Notes: (A1)(ft) for correct fractions from their Venn diagram
(M1) for multiplying their fractions
(A1)(ft) for correct answer.
[3 marks]
\(\frac{6}{{15}}{\text{ }}\left( {\frac{2}{5}{\text{, }}0.4{\text{, }}40\% } \right)\) (A1)(ft)(A1)(ft)
Note: (A1)(ft) for numerator, (A1)(ft) denominator.
[2 marks]
(A1)(A1)(A1)
Note: (A1) for \(0.8\), (A1) for \(0.7\), (A1) for \(0.6\) and \(0.4\).
[3 marks]
(i) \(0.2 \times 0.7 = 0.14\) (M1)(A1)(ft)(G2)
Note: (M1) for multiplying correct numbers.
[2 marks]
(ii) \(0.2 \times 0.3 + 0.8 \times 0.6\) (M1)(M1)
\( = 0.54\) (A1)(ft)(G3)
Note: (M1) for each correct product (use candidate’s tree), (A1)(ft) for answer.
[3 marks]
Examiners report
The Venn diagram was well drawn on the whole although some of the candidates missed out the Universal box and others filled in the intersections wrongly but still gained ft marks for the remaining parts of the question.
Well answered.
Well answered.
Few correct answers. Either candidates added instead of multiplying or they used replacement and so the fractions given were the same.
Again few correct answers. Candidates wrote the answer out of \(50\) instead of \(15\).
The tree diagram was well done on the whole. It appears as if some candidates may have completed this on the exam paper and this was not included with their papers. However, the question did state clearly “Copy and complete …”
(i) This part was well done by those candidates who remembered to multiply instead of add.
(ii) Many candidates just wrote down “Claire” for this answer. Others wrongly multiplied or added \(0.3\) with \(0.6\).
