| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2012 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Principle of Inclusion/Exclusion |
| Type | Standard Survey to Venn Diagram |
| Difficulty | Moderate -0.8 This is a straightforward application of inclusion-exclusion to construct a Venn diagram with three sets, followed by routine probability calculations. The question provides all necessary information explicitly, requires only systematic arithmetic to fill in the diagram regions, and the probability calculations are direct counting exercises. This is easier than average as it's a standard textbook exercise with no conceptual challenges or problem-solving required. |
| Spec | 2.03b Probability diagrams: tree, Venn, sample space |
| Answer | Marks | Guidance |
|---|---|---|
| (a) 3 closed curves and 25 in correct place 15,10,5 or 15,3,20 | M1 A1 A1 | Labels \(R, S, C\) and box |
| (4 marks) | ||
| (b) All values/100 or equivalent fractions award accuracy marks. 7/100 or 0.07 | M1 A1 | M1 for 'their 7'/100 seen in diagram or here)/100. A1 Correct answer only |
| (2 marks) | ||
| (c) \((3+5)/100 = 2/25\) or 0.08 | M1 A1 | For ('their 3'+'their 5')/100. \(\frac{8}{48}\) award M0. A1 Correct answer only or equivalent. |
| (2 marks) | ||
| (d) \((25+15+10+5)/100 = 11/20\) or 0.55 | M1 A1 | M1 Accept sum of their 4 values from the Venn diagram /100. A1 Correct answer only or equivalent |
| (2 marks) | ||
| (e) \(P(S \cap C' | R) = \frac{P(S \cap C' \cap R)}{P(R)}\) | M1 |
| \(= \frac{15}{65}\) | A1 | require 'their 15' and correct denominator of 65 |
| \(= \frac{3}{13}\) | A1 | A1 for exact equivalent answers, including 15/65. In all parts correct answers with no working award full marks. |
| (3 marks) |
**(a)** 3 closed curves and 25 in correct place 15,10,5 or 15,3,20 | M1 A1 A1 | Labels $R, S, C$ and box
| | (4 marks) |
**(b)** All values/100 or equivalent fractions award accuracy marks. 7/100 or 0.07 | M1 A1 | M1 for 'their 7'/100 seen in diagram or here)/100. A1 Correct answer only
| | (2 marks) |
**(c)** $(3+5)/100 = 2/25$ or 0.08 | M1 A1 | For ('their 3'+'their 5')/100. $\frac{8}{48}$ award M0. A1 Correct answer only or equivalent.
| | (2 marks) |
**(d)** $(25+15+10+5)/100 = 11/20$ or 0.55 | M1 A1 | M1 Accept sum of their 4 values from the Venn diagram /100. A1 Correct answer only or equivalent
| | (2 marks) |
**(e)** $P(S \cap C'|R) = \frac{P(S \cap C' \cap R)}{P(R)}$ | M1 | Require denominator to be 'their 65' or 'their $\frac{65}{100}$'.
| | |
| $= \frac{15}{65}$ | A1 | require 'their 15' and correct denominator of 65
| | |
| $= \frac{3}{13}$ | A1 | A1 for exact equivalent answers, including 15/65. In all parts correct answers with no working award full marks.
| | (3 marks) |
**Total: 13 marks**
---\begin{enumerate}
\item The following shows the results of a survey on the types of exercise taken by a group of 100 people.
\end{enumerate}
65 run\\
48 swim\\
60 cycle\\
40 run and swim\\
30 swim and cycle\\
35 run and cycle\\
25 do all three\\
(a) Draw a Venn Diagram to represent these data.
Find the probability that a randomly selected person from the survey\\
(b) takes none of these types of exercise,\\
(c) swims but does not run,\\
(d) takes at least two of these types of exercise.
Jason is one of the above group.\\
Given that Jason runs,\\
(e) find the probability that he swims but does not cycle.\\
\hfill \mbox{\textit{Edexcel S1 2012 Q6 [13]}}