| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2005 |
| Session | June |
| Marks | 19 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | Direct binomial probability calculation |
| Difficulty | Moderate -0.3 This is a straightforward S1 binomial distribution question requiring direct application of formulas with calculator use. Parts (a) and (b) are standard probability calculations using B(n,p), part (c) involves routine manipulation of mean and variance formulas (np and np(1-p)). All techniques are textbook exercises with no problem-solving insight required, though the multi-part structure and cumulative probability calculations place it slightly below average difficulty rather than being trivial. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04d Normal approximation to binomial |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(B(n, 0.07)\) | M1 | Use of in (a) |
| \(P(X=2) = \binom{17}{2}(0.07)^2(0.93)^{15}\) | A1 | Fully correct expression. May be implied |
| \(= 136 \times 0.0049 \times 0.33670\) | ||
| \(= 0.224\) to \(0.225\) | A1 | AWFW (0.22438) |
| Total: 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(X \leq 5 \mid B(50, 0.07))\) | M1 | Attempted; tables or formula (\(\geq 3\) terms stated). May be implied |
| \(= 0.865\) | A1 | AWRT (0.8650) |
| Total: 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(B(50, 0.55)\) | ||
| \(P(Y \geq 30) = P(Y' \leq 20)\) | M1 | Change from \(Y\) to \(Y'\). Must be clear evidence |
| with \(p = 0.45\) | A1 | Stated or implied |
| \(= 0.286\) | A1 | AWRT (0.2862) |
| Total: 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Estimate of \(p = \dfrac{10}{50} = 0.2\) | B1 | CAO |
| Total: 1 mark |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Estimate of \(\text{SD}(X) = \sqrt{np(1-p)}\) | M1 | Use of; accept no \(\sqrt{\phantom{x}}\) |
| \(= \sqrt{50 \times 0.2 \times 0.8} = \sqrt{8}\) | ||
| \(= 2.82\) to \(2.83\) | A1 | AWFW; accept \(\sqrt{8}\) |
| Total: 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\text{SD}(X)\) less than \(6.8\) or \(V(X)\) less than \(46.24\) | M1\(\checkmark\) | Comparison \(\checkmark\) on (c)(ii). Must be like with like |
| Not a reasonable assumption | A1\(\checkmark\) | \(\checkmark\) on (c)(ii) and like with like comparison |
| Total: 2 marks |
# Question 5:
## Part (a)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $B(n, 0.07)$ | M1 | Use of in (a) |
| $P(X=2) = \binom{17}{2}(0.07)^2(0.93)^{15}$ | A1 | Fully correct expression. May be implied |
| $= 136 \times 0.0049 \times 0.33670$ | | |
| $= 0.224$ to $0.225$ | A1 | AWFW (0.22438) |
| **Total: 3 marks** | | |
## Part (a)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(X \leq 5 \mid B(50, 0.07))$ | M1 | Attempted; tables or formula ($\geq 3$ terms stated). May be implied |
| $= 0.865$ | A1 | AWRT (0.8650) |
| **Total: 2 marks** | | |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $B(50, 0.55)$ | | |
| $P(Y \geq 30) = P(Y' \leq 20)$ | M1 | Change from $Y$ to $Y'$. Must be clear evidence |
| with $p = 0.45$ | A1 | Stated or implied |
| $= 0.286$ | A1 | AWRT (0.2862) |
| **Total: 3 marks** | | |
## Part (c)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| Estimate of $p = \dfrac{10}{50} = 0.2$ | B1 | CAO |
| **Total: 1 mark** | | |
## Part (c)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Estimate of $\text{SD}(X) = \sqrt{np(1-p)}$ | M1 | Use of; accept no $\sqrt{\phantom{x}}$ |
| $= \sqrt{50 \times 0.2 \times 0.8} = \sqrt{8}$ | | |
| $= 2.82$ to $2.83$ | A1 | AWFW; accept $\sqrt{8}$ |
| **Total: 2 marks** | | |
## Part (c)(iii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{SD}(X)$ less than $6.8$ or $V(X)$ less than $46.24$ | M1$\checkmark$ | Comparison $\checkmark$ on (c)(ii). Must be like with like |
| **Not** a reasonable assumption | A1$\checkmark$ | $\checkmark$ on (c)(ii) and like with like comparison |
| **Total: 2 marks** | | |
---5
\begin{enumerate}[label=(\alph*)]
\item At a particular checkout in a supermarket, the probability that the barcode reader fails to read the barcode first time on any item is 0.07 , and is independent from item to item.
\begin{enumerate}[label=(\roman*)]
\item Calculate the probability that, from a shopping trolley containing 17 items, the reader fails to read the barcode first time on exactly 2 of the items.
\item Determine the probability that, from a shopping trolley containing 50 items, the reader fails to read the barcode first time on at most 5 of the items.
\end{enumerate}\item At another checkout in the supermarket, the probability that a faulty barcode reader fails to read the barcode first time on any item is 0.55 , and is independent from item to item.
Determine the probability that, from a shopping trolley containing 50 items, this reader fails to read the barcode first time on at least 30 of the items.
\item At a third checkout in the supermarket, a record is kept of $X$, the number of times per 50 items that the barcode reader fails to read a barcode first time. An analysis of the records gives a mean of 10 and a standard deviation of 6.8.
\begin{enumerate}[label=(\roman*)]
\item Estimate $p$, the probability that the barcode reader fails to read a barcode first time.
\item Using your estimate of $p$ and assuming that $X$ can be modelled by a binomial distribution, estimate the standard deviation of $X$.
\item Hence comment on the assumption that $X$ can be modelled by a binomial distribution.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S1 2005 Q5 [19]}}