| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Session | Specimen |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Principle of Inclusion/Exclusion |
| Type | Three-Set Venn Diagram Probability Calculation |
| Difficulty | Moderate -0.8 This is a straightforward Venn diagram reading exercise requiring only basic probability calculations (counting regions and dividing by totals). All information is explicitly given in the diagram, requiring no problem-solving insight—just careful arithmetic and recall of independence definition. Significantly easier than average A-level questions. |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{2+3}{\text{their total}} = \frac{5}{\text{their total}} = \frac{1}{6}\) (given answer) | M1 A1cso | M1 for \(\frac{2+3}{\text{their total}}\) or \(\frac{5}{30}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{4+2+5+3}{\text{total}} = \frac{14}{30}\) or \(\frac{7}{15}\) or 0.4\(\dot{6}\) | M1 A1 | M1 for adding at least 3 of "4,2,5,3" and dividing by total |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(A \cap C) = 0\) | B1 | B1 for 0 or 0/30 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(C \text{ reads at least one magazine}) = \frac{6+3}{20} = \frac{9}{20}\) | M1 A1 | M1 for denominator of 20 or \(\frac{20}{30}\) leading to denominator 20 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(B) = \frac{10}{30} = \frac{1}{3}\), \(P(C) = \frac{9}{30} = \frac{3}{10}\), \(P(B \cap C) = \frac{3}{30} = \frac{1}{10}\) | M1 | 1st M1 for attempting all required probabilities |
| \(P(B)\times P(C) = \frac{1}{3}\times\frac{3}{10} = \frac{1}{10} = P(B \cap C)\) or \(P(B | C) = \frac{3}{9} = \frac{1}{3} = P(B)\) | M1 |
| So yes they are statistically independent | A1cso | A1 for fully correct test with comment |
# Question 4:
## Part (a)
| $\frac{2+3}{\text{their total}} = \frac{5}{\text{their total}} = \frac{1}{6}$ (**given answer**) | M1 A1cso | M1 for $\frac{2+3}{\text{their total}}$ or $\frac{5}{30}$ |
## Part (b)
| $\frac{4+2+5+3}{\text{total}} = \frac{14}{30}$ or $\frac{7}{15}$ or 0.4$\dot{6}$ | M1 A1 | M1 for adding at least 3 of "4,2,5,3" and dividing by total |
## Part (c)
| $P(A \cap C) = 0$ | B1 | B1 for 0 or 0/30 |
## Part (d)
| $P(C \text{ reads at least one magazine}) = \frac{6+3}{20} = \frac{9}{20}$ | M1 A1 | M1 for denominator of 20 or $\frac{20}{30}$ leading to denominator 20 |
## Part (e)
| $P(B) = \frac{10}{30} = \frac{1}{3}$, $P(C) = \frac{9}{30} = \frac{3}{10}$, $P(B \cap C) = \frac{3}{30} = \frac{1}{10}$ | M1 | 1st M1 for attempting all required probabilities |
|---|---|---|
| $P(B)\times P(C) = \frac{1}{3}\times\frac{3}{10} = \frac{1}{10} = P(B \cap C)$ or $P(B|C) = \frac{3}{9} = \frac{1}{3} = P(B)$ | M1 | 2nd M1 for correct test with all correct probabilities |
| So yes they are statistically independent | A1cso | A1 for fully correct test with comment |
---\begin{enumerate}
\item The Venn diagram in Figure 1 shows the number of students in a class who read any of 3 popular magazines $A , B$ and $C$.
\end{enumerate}
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}
% Outer rectangle
\draw[thick] (-4,-2.5) rectangle (4.5,2.8);
% Circle A (upper left)
\draw[thick] (-1.8,0.5) ellipse (2 and 1);
\node[above left] at (-2.2,1.7) {\textit{A}};
% Circle B (upper center-right, larger)
\draw[thick] (1.2,0.4) ellipse (2 and 1);
\node[above] at (1.2,1.6) {\textit{B}};
% Ellipse C (lower right)
\draw[thick] (2.2,-0.5) ellipse (2 and 1);
\node[right] at (3.8,-0.0) {\textit{C}};
% Region labels
\node at (-2.2,0.5) {4};
\node at (-0.5,0.7) {2};
\node at (1.2,1.2) {5};
\node at (1.8,-0.2) {3};
\node at (3.0,-0.7) {6};
\node at (-2.8,-1.8) {10};
\end{tikzpicture}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
One of these students is selected at random.\\
(a) Show that the probability that the student reads more than one magazine is $\frac { 1 } { 6 }$.\\
(b) Find the probability that the student reads $A$ or $B$ (or both).\\
(c) Write down the probability that the student reads both $A$ and $C$.
Given that the student reads at least one of the magazines,\\
(d) find the probability that the student reads $C$.\\
(e) Determine whether or not reading magazine $B$ and reading magazine $C$ are statistically independent.\\
\hfill \mbox{\textit{Edexcel S1 Q4 [10]}}