June 2023 Paper 3 Q12
12 It 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 \(\mathrm{B}(32, 0.4)\)
(a) State one assumption, in context, required for this distribution to be used. [1 mark]
(b) Find the probability that exactly 7 of the drivers on a particular day pass their test. [1 mark]
(c) Find the probability that, at most, 16 of the drivers on a particular day pass their test. [1 mark]
(d) Find the probability that more than 12 of the drivers on a particular day pass their test. [2 marks]
(e) Find the mean number of drivers per day who pass their test. [1 mark]
(f) Find the standard deviation of the number of drivers per day who pass their test. [2 marks]
| Scheme | Marks | AO |
|---|---|---|
States one of the following assumptions in context
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 | E1 | 3.5b |
| (1) |
Typical solution
The probability of passing the driving test is constant
| Scheme | Marks | AO |
|---|---|---|
| Obtains correct probability AWFW [0.0156, 0.016] | B1 | 1.1b |
| (1) |
Typical solution
0.0157
| Scheme | Marks | AO |
|---|---|---|
| Obtains correct probability AWFW [0.908, 0.91] | B1 | 1.1b |
| (1) |
Typical solution
0.908
| Scheme | Marks | AO |
|---|---|---|
| States \(\mathrm{P}(X \geqslant 13)\) or \(\mathrm{P}(13 \leqslant X \leqslant 32)\) or \(1 - \mathrm{P}(X \leqslant 12)\) or \(1 -\) [0.46, 0.462] PI by correct answer | M1 | 1.1a |
| Obtains correct probability AWFW [0.538, 0.54] | A1 | 1.1b |
| (2) |
Typical solution
\[\begin{aligned}\mathrm{P}(X \gt 12) &= 1 - \mathrm{P}(X \leqslant 12) \\ &= 1 - 0.4618 \\ &= 0.538\end{aligned}\]| Scheme | Marks | AO |
|---|---|---|
| Obtains 12.8 Do not ISW | B1 | 1.1b |
| (1) |
Typical solution
12.8
| Scheme | Marks | AO |
|---|---|---|
| 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] or \(\dfrac{8\sqrt{3}}{5}\) Ignore incorrect labels Condone missing brackets | M1 | 1.1a |
| Obtains the correct standard deviation AWFW [2.77, 2.8] or \(\dfrac{8\sqrt{3}}{5}\) Do not ignore incorrect labels Do not ISW Do not allow \(\sqrt{7.68}\) | A1 | 1.1b |
| (2) | ||
| (8 marks) |