| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2007 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Direct binomial probability calculation |
| Difficulty | Moderate -0.8 This is a straightforward application of binomial distribution formulas with clearly stated parameters. Parts (a) and (b) require direct use of P(X=r) and cumulative probability calculations, while part (c) involves standard mean/variance formulas. No problem-solving insight needed—purely computational with standard S1 techniques. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04d Normal approximation to binomial |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Use of binomial in (a), (b) or (c) | M1 | Can be implied |
| \(P(E=5) = \binom{16}{5}(p)^5(1-p)^{11}\) | M1 | Allow \(p = 0.45, 0.25, 0.30\) or \(\frac{1}{3}\) |
| \(= 0.112\) | A1 | AWRT (0.1123) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(B(50, 0.25)\) | B1 | Used; can be implied |
| \(P(C \leq 12) = 0.511\) | B1 | AWRT (0.5110) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P(10 < B' < 20) = 0.9152\) or \(0.9522\) | M1 | Allow 3 dp accuracy |
| minus \(0.0789\) or \(0.1390\) | M1 | Allow 3 dp accuracy |
| \(= 0.836\) | A1 | AWRT (0.8363) |
| or \(B(50, 0.30)\) expressions stated for at least 3 terms within \(10 \leq B' \leq 20\); Answer \(= 0.836\) | (M1)(A2) | Or implied by a correct answer; AWRT |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(n = 40\), \(p = 0.7\) | B1 | Both used; can be implied |
| Mean \(\mu = np = 28\) | B1\(\sqrt{}\) | CAO; \(\sqrt{}\) on \(p\) only |
| Variance \(\sigma^2 = np(1-p) = 8.4\) | M1 | Use of \(np(1-p)\) even if SD |
| Standard deviation \(= \sqrt{8.4} = 2.89\) to \(2.9\) | A1 | CAO; AWFW |
# Question 2:
## Part 2(a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Use of binomial in (a), (b) or (c) | M1 | Can be implied |
| $P(E=5) = \binom{16}{5}(p)^5(1-p)^{11}$ | M1 | Allow $p = 0.45, 0.25, 0.30$ or $\frac{1}{3}$ |
| $= 0.112$ | A1 | AWRT (0.1123) |
## Part 2(b)(i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $B(50, 0.25)$ | B1 | Used; can be implied |
| $P(C \leq 12) = 0.511$ | B1 | AWRT (0.5110) |
## Part 2(b)(ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P(10 < B' < 20) = 0.9152$ or $0.9522$ | M1 | Allow 3 dp accuracy |
| minus $0.0789$ or $0.1390$ | M1 | Allow 3 dp accuracy |
| $= 0.836$ | A1 | AWRT (0.8363) |
| **or** $B(50, 0.30)$ expressions stated for **at least 3** terms within $10 \leq B' \leq 20$; Answer $= 0.836$ | (M1)(A2) | Or implied by a correct answer; AWRT |
## Part 2(c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $n = 40$, $p = 0.7$ | B1 | Both used; can be implied |
| Mean $\mu = np = 28$ | B1$\sqrt{}$ | CAO; $\sqrt{}$ on $p$ only |
| Variance $\sigma^2 = np(1-p) = 8.4$ | M1 | Use of $np(1-p)$ even if SD |
| Standard deviation $= \sqrt{8.4} = 2.89$ to $2.9$ | A1 | CAO; AWFW |
---2 A hotel has 50 single rooms, 16 of which are on the ground floor. The hotel offers guests a choice of a full English breakfast, a continental breakfast or no breakfast. The probabilities of these choices being made are $0.45,0.25$ and 0.30 respectively. It may be assumed that the choice of breakfast is independent from guest to guest.
\begin{enumerate}[label=(\alph*)]
\item On a particular morning there are 16 guests, each occupying a single room on the ground floor. Calculate the probability that exactly 5 of these guests require a full English breakfast.
\item On a particular morning when there are 50 guests, each occupying a single room, determine the probability that:
\begin{enumerate}[label=(\roman*)]
\item at most 12 of these guests require a continental breakfast;
\item more than 10 but fewer than 20 of these guests require no breakfast.
\end{enumerate}\item When there are 40 guests, each occupying a single room, calculate the mean and the standard deviation for the number of these guests requiring breakfast.
\end{enumerate}
\hfill \mbox{\textit{AQA S1 2007 Q2 [12]}}