| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Principle of Inclusion/Exclusion |
| Type | Range of Possible Values |
| Difficulty | Moderate -0.8 Part (a) requires basic set theory reasoning with inclusion-exclusion to establish inequalities (12 ≤ x ≤ 33), which is straightforward once the constraints are identified. Part (b) tests understanding of independence by checking if P(A)P(M) = P(A∩M), requiring simple probability calculations. This is a standard AS-level application question with clear structure and no novel insight required, making it easier than average. |
| Spec | 2.03b Probability diagrams: tree, Venn, sample space2.03c Conditional probability: using diagrams/tables |
| Answer | Marks | Guidance |
|---|---|---|
| \(45 - 25 = 20\), or e.g. '25, 13+12+y, 45' giving \(12, x, 32\) | M1 | Attempting to find range for \(x\) or attempt to find largest/smallest number of students that could study Music only (may be implied by one correct endpoint) |
| \(12 \leq x \leq 32\) | A1 | Allow \(12-32\) or \(x\ldots12\) and \(x\ldots32\). Note: \(12 < x < 32\) scores M1A0 |
| Answer | Marks | Guidance |
|---|---|---|
| To be independent: \(P(A) \times P(M) = P(A \text{ and } M)\) | M1 | Must use \(A\) and \(M\); allow any rearrangement; allow all three probabilities labelled followed by correct equation |
| Answer | Marks | Guidance |
|---|---|---|
| or \(\frac{25}{45} \times \frac{x}{45} = \frac{12}{45}\) or \(\frac{25}{45} \times \frac{12+y}{45} = \frac{12}{45}\) | A1 | \(P(M) = 0.48\) or correct equation for \(P(M)\), or \(x\) or \(y\). Do not award if working with numbers e.g. \(P(A \text{ and } M) = 12\) |
| The number of students taking part in music would be \(\frac{12}{25} \times 45 = 21.6\) | A1 | Dependent on M1 only; 21.6 or \(\frac{21.6}{45}\); or \(y = 9.6\) |
| ...so it is not possible for \(A\) and \(M\) to be independent (since it must be a whole number) | A1 | Dependent on all previous marks; correct deduction from correct working. Ignore any reference to the range of values found in part (a). |
## Question 3:
**Part (a):**
$45 - 25 = 20$, or e.g. '25, 13+12+y, 45' giving $12, x, 32$ | M1 | Attempting to find range for $x$ or attempt to find largest/smallest number of students that could study Music only (may be implied by one correct endpoint)
$12 \leq x \leq 32$ | A1 | Allow $12-32$ or $x\ldots12$ and $x\ldots32$. Note: $12 < x < 32$ scores M1A0
**Part (b):**
To be independent: $P(A) \times P(M) = P(A \text{ and } M)$ | M1 | Must use $A$ and $M$; allow any rearrangement; allow all three probabilities labelled followed by correct equation
$$P(M) = \frac{P(A \text{ and } M)}{P(A)} = \frac{\frac{12}{45}}{\frac{25}{45}} = \frac{12}{25}$$
or $\frac{25}{45} \times \frac{x}{45} = \frac{12}{45}$ or $\frac{25}{45} \times \frac{12+y}{45} = \frac{12}{45}$ | A1 | $P(M) = 0.48$ or correct equation for $P(M)$, or $x$ or $y$. Do not award if working with numbers e.g. $P(A \text{ and } M) = 12$
The number of students taking part in music would be $\frac{12}{25} \times 45 = 21.6$ | A1 | Dependent on M1 only; 21.6 or $\frac{21.6}{45}$; or $y = 9.6$
...so it is not possible for $A$ and $M$ to be independent (since it must be a whole number) | A1 | Dependent on all previous marks; correct deduction from correct working. Ignore any reference to the range of values found in part (a).
**SC:** If M0 scored, allow access to 1st and 2nd A1 (maximum M0A1A1A0)
---3. In an after-school club, students can choose to take part in Art, Music, both or neither.
There are 45 students that attend the after-school club. Of these
\begin{itemize}
\item 25 students take part in Art
\item 12 students take part in both Art and Music
\item the number of students that take part in Music is $x$
\begin{enumerate}[label=(\alph*)]
\item Find the range of possible values of $x$
\end{itemize}
One of the 45 students is selected at random.\\
Event $A$ is the event that the student selected takes part in Art.\\
Event $M$ is the event that the student selected takes part in Music.
\item Determine whether or not it is possible for the events $A$ and $M$ to be independent.
\end{enumerate}
\hfill \mbox{\textit{Edexcel AS Paper 2 2023 Q3 [6]}}