| Exam Board | AQA |
|---|---|
| Module | Paper 3 (Paper 3) |
| Year | 2023 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | E(X) and Var(X) with probability calculations |
| Difficulty | Easy -1.3 This is a straightforward binomial distribution question requiring only standard recall and calculator work. Parts (a)-(f) involve stating assumptions, direct application of binomial probability formulas (likely using calculator functions), and recalling mean/variance formulas for B(n,p). No problem-solving, proof, or conceptual insight is needed—just routine application of well-practiced techniques. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| Answer | Marks |
|---|---|
| 12(a) | States one of the following |
| Answer | Marks | Guidance |
|---|---|---|
| drivers or tests | 3.5b | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| Subtotal | 1 | |
| Q | Marking instructions | AO |
| Answer | Marks | Guidance |
|---|---|---|
| 12(b) | Obtains correct probability | |
| AWFW [0.0156, 0.016] | 1.1b | B1 |
| Subtotal | 1 | |
| Q | Marking instructions | AO |
| Answer | Marks | Guidance |
|---|---|---|
| 12(c) | Obtains correct probability | |
| AWFW [0.908, 0.91] | 1.1b | B1 |
| Subtotal | 1 | |
| Q | Marking instructions | AO |
| Answer | Marks |
|---|---|
| 12(d) | States P(X ≥ 13) or |
| Answer | Marks | Guidance |
|---|---|---|
| PI by correct answer | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| AWFW [0.538, 0.54] | 1.1b | A1 |
| Subtotal | 2 | |
| Q | Marking instructions | AO |
| Answer | Marks | Guidance |
|---|---|---|
| 12(e) | Obtains 12.8 | |
| Do not ISW | 1.1b | B1 |
| Subtotal | 1 | |
| Q | Marking instructions | AO |
| Answer | Marks |
|---|---|
| 12(f) | Uses the correct formula for |
| Answer | Marks | Guidance |
|---|---|---|
| Condone missing brackets | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Do not allow 7.68 | 1.1b | A1 |
| Subtotal | 2 | |
| Question 12 Total | 8 | |
| Q | Marking instructions | AO |
Question 12:
--- 12(a) ---
12(a) | States one of the following
assumptions in context
• probability of passing the
test is constant or 0.4 or
fixed or the same or does
not change
• passing the test occurs
independently
• only two outcomes of
passing or failing test
Do not ignore incorrect
statements about any of the
above
Must use ‘test’
Condone 'exam' for ‘test’
Allow equivalent statements for
failing for the reference to
probability or independence
Do not allow probability being
independent
Do not allow fixed number of
drivers or tests | 3.5b | E1 | The probability of passing the
driving test is constant
Subtotal | 1
Q | Marking instructions | AO | Marks | Typical solution
--- 12(b) ---
12(b) | Obtains correct probability
AWFW [0.0156, 0.016] | 1.1b | B1 | 0.0157
Subtotal | 1
Q | Marking instructions | AO | Marks | Typical solution
--- 12(c) ---
12(c) | Obtains correct probability
AWFW [0.908, 0.91] | 1.1b | B1 | 0.908
Subtotal | 1
Q | Marking instructions | AO | Marks | Typical solution
--- 12(d) ---
12(d) | States P(X ≥ 13) or
P(13 ≤ X ≤ 32) or 1 – P(X ≤12)
or 1 – [0.46, 0.462]
PI by correct answer | 1.1a | M1 | P(X > 12) = 1 – P(X ≤12)
= 1 – 0.4618
= 0.538
Obtains correct probability
AWFW [0.538, 0.54] | 1.1b | A1
Subtotal | 2
Q | Marking instructions | AO | Marks | Typical solution
--- 12(e) ---
12(e) | Obtains 12.8
Do not ISW | 1.1b | B1 | 12.8
Subtotal | 1
Q | Marking instructions | AO | Marks | Typical solution
--- 12(f) ---
12(f) | Uses the correct formula for
variance or standard deviation
with 32, 0.4 and 0.6 substituted
OE
PI by 7.68 or AWFW [2.77, 2.8]
8 3
or
5
Ignore incorrect labels
Condone missing brackets | 1.1a | M1 | Variance = 32 × 0.4 × 0.6 = 7.68
Standard deviation = 2.77
Obtains the correct standard
deviation
8 3
AWFW [2.77, 2.8] or
5
Do not ignore incorrect labels
Do not ISW
Do not allow 7.68 | 1.1b | A1
Subtotal | 2
Question 12 Total | 8
Q | Marking instructions | AO | Marks | Typical solutionIt is known that, on average, 40% of the drivers who take their driving test at a local test centre pass their driving test.
Each day 32 drivers take their driving test at this centre.
The number of drivers who pass their test on a particular day can be modelled by the distribution B $(32, 0.4)$
\begin{enumerate}[label=(\alph*)]
\item State one assumption, in context, required for this distribution to be used.
[1 mark]
\item Find the probability that exactly 7 of the drivers on a particular day pass their test.
[1 mark]
\item Find the probability that, at most, 16 of the drivers on a particular day pass their test.
[1 mark]
\item Find the probability that more than 12 of the drivers on a particular day pass their test.
[2 marks]
\item Find the mean number of drivers per day who pass their test.
[1 mark]
\item Find the standard deviation of the number of drivers per day who pass their test.
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Paper 3 2023 Q12 [8]}}