| Exam Board | AQA |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2024 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Sum or product of two independent values |
| Difficulty | Moderate -0.3 This is a straightforward probability distribution question requiring basic probability axioms (probabilities sum to 1), enumeration of simple cases for two independent events, and standard assumptions about independence. All techniques are routine for AS-level statistics with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04a Discrete probability distributions |
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 or more |
| P(\(X = x\)) | 0.03 | 0.15 | 0.22 | 0.31 | 0.09 | \(p\) |
| Answer | Marks | Guidance |
|---|---|---|
| 15(a) | Sums probabilities and equates | |
| to one PI | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Obtains p = 0.2 | 1.1b | A1 |
| Subtotal | 2 | |
| Q | Marking instructions | AO |
| Answer | Marks | Guidance |
|---|---|---|
| 15(b) | Obtains at least one numerically | |
| correct term ACF | 3.1b | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| added to the first term | 3.1b | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| OE | 1.1b | A1 |
| Subtotal | 3 | |
| Q | Marking instructions | AO |
| Answer | Marks |
|---|---|
| 15(c)(i) | Identifies independence |
| Answer | Marks | Guidance |
|---|---|---|
| probabilities staying the same | 3.2b | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| Subtotal | 1 | |
| Q | Marking instructions | AO |
| Answer | Marks |
|---|---|
| 15(c)(ii) | Allow argument both ways with |
| Answer | Marks | Guidance |
|---|---|---|
| representative of the population | 2.2b | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| Subtotal | 1 | |
| Question 15 Total | 7 | |
| Q | Marking instructions | AO |
Question 15:
--- 15(a) ---
15(a) | Sums probabilities and equates
to one PI | 1.1a | M1 | 0.03+0.15+0.22+0.31+0.09+p = 1
p = 1 – (0.03+0.15+0.22+0.31+0.09)
= 0.20
Obtains p = 0.2 | 1.1b | A1
Subtotal | 2
Q | Marking instructions | AO | Marks | Typical solution
--- 15(b) ---
15(b) | Obtains at least one numerically
correct term ACF | 3.1b | M1 | P(total is 3)
= P(2 ∩ 1) + P(1 ∩ 2) + P(0 ∩ 3) + P(3
∩ 0)
= 0.22×0.15 + 0.15×0.22 + 0.03×0.31
+ 0.31×0.03
= 2(0.033) + 2(0.0093)
= 0.0846
Obtains a correct numerically
different second term which is
added to the first term | 3.1b | M1
Obtains 0.0846
OE | 1.1b | A1
Subtotal | 3
Q | Marking instructions | AO | Marks | Typical solution
--- 15(c)(i) ---
15(c)(i) | Identifies independence
or
defines independence in context
Or
Identifies that the plants must be
representative of the population.
Do not accept assumption
probabilities staying the same | 3.2b | E1 | The number of flowers that grow on
one plant must be independent of the
number that grow on any other plant
Subtotal | 1
Q | Marking instructions | AO | Marks | Typical solution
--- 15(c)(ii) ---
15(c)(ii) | Allow argument both ways with
a valid contextualised reason
from their assumption of
independence from (c)(i)
E.g. Could be independent as
the plants are chosen randomly
from a large batch, not
dependent on each other for
production of flowers
Dependent because grown in
the same soil, compete for
resources, same batch of seeds.
Or
As plants are from a large batch,
they are likely to be
representative of the population | 2.2b | E1 | The plants may be grown in identical
conditions and therefore the number of
flowers that grow on each plant may
not be independent.
Subtotal | 1
Question 15 Total | 7
Q | Marking instructions | AO | Marks | Typical solutionThe number of flowers which grow on a certain type of plant can be modelled by the discrete random variable $X$
The probability distribution of $X$ is given in the table below.
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$x$ & 0 & 1 & 2 & 3 & 4 & 5 or more \\
\hline
P($X = x$) & 0.03 & 0.15 & 0.22 & 0.31 & 0.09 & $p$ \\
\hline
\end{tabular}
\begin{enumerate}[label=(\alph*)]
\item Find the value of $p$
[2 marks]
\item Two plants of this type are randomly selected from a large batch received from a local garden centre.
Find the probability that the two plants will produce a total of three flowers.
[3 marks]
\item \begin{enumerate}[label=(\roman*)]
\item State one assumption necessary for the calculation in part (b) to be valid.
[1 mark]
\item Comment on whether, in reality, this assumption is likely to be valid.
[1 mark]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{AQA AS Paper 2 2024 Q15 [7]}}