| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | October |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Sampling distribution of mean or linear combination |
| Difficulty | Standard +0.3 Part (a) requires systematic enumeration of outcomes for two customers and calculating probabilities using given ratios—straightforward but tedious. Part (b) involves finding the mean number of scoops and applying normal approximation with continuity correction, which is a standard S2 technique. The question is slightly easier than average as it follows predictable patterns for sampling distribution problems, though the calculation work is non-trivial. |
| Spec | 2.04b Binomial distribution: as model B(n,p)5.01a Permutations and combinations: evaluate probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Possible values of \(S = 2, 3, 4, 5, 6\) | B1 | B1 for all 5 values of \(S\) |
| \(P(S=2) = \frac{5}{8} \times \frac{5}{8}\) and \(P(S=6) = \frac{1}{8} \times \frac{1}{8}\) | M1 A1 | 1st M1 for \(p^2\) for \(P(S=2)\) or \(P(S=6)\); 1st A1 for \(\frac{25}{64}\) and \(\frac{1}{64}\) (allow awrt 0.391 and awrt 0.0156) |
| \(P(S=3) = 2 \times \frac{5}{8} \times \frac{2}{8}\), \(P(S=4) = \frac{2}{8} \times \frac{2}{8} + 2 \times \frac{5}{8} \times \frac{1}{8}\), \(P(S=5) = 2 \times \frac{1}{8} \times \frac{2}{8}\) | M1 | 2nd M1 for \(2pq\) for \(P(S=3)\) or \(P(S=5)\), or \(p^2 + 2pq\) for \(P(S=4)\) |
| Complete table: \(P(S=2)=\frac{25}{64}\), \(P(S=3)=\frac{20}{64}\), \(P(S=4)=\frac{14}{64}\), \(P(S=5)=\frac{4}{64}\), \(P(S=6)=\frac{1}{64}\) | A1 | 2nd A1 for complete correct probability distribution (allow awrt 3sf); Note: if 5 values of \(S\) not found, can score B0M1A1M1A0 |
| (5) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Let \(X\) = number of scoops ordered by \(n\) customers; \(P(X=n) = \left(\frac{5}{8}\right)^n\) | ||
| \(P(X > n) = 1 - \left(\frac{5}{8}\right)^n\) | M1 | 1st M1 for \(1 - p^n > 0.99\) (allow equation) or equivalent e.g. \(p^n < 0.01\) |
| \(1 - \left(\frac{5}{8}\right)^n > 0.99\) | ||
| \(\left(\frac{5}{8}\right)^n < 0.01\) | dM1 | 2nd dM1 dependent on 1st M1 for correct use of a valid method to solve to \(n > k\) (allow equation) |
| \(n > 9.798\ldots\) | ||
| \(n = 10\) | A1cao | A1 for 10 cao |
| (3) | ||
| [8 marks] |
# Question 7:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Possible values of $S = 2, 3, 4, 5, 6$ | **B1** | B1 for all 5 values of $S$ |
| $P(S=2) = \frac{5}{8} \times \frac{5}{8}$ and $P(S=6) = \frac{1}{8} \times \frac{1}{8}$ | **M1 A1** | 1st M1 for $p^2$ for $P(S=2)$ or $P(S=6)$; 1st A1 for $\frac{25}{64}$ and $\frac{1}{64}$ (allow awrt 0.391 and awrt 0.0156) |
| $P(S=3) = 2 \times \frac{5}{8} \times \frac{2}{8}$, $P(S=4) = \frac{2}{8} \times \frac{2}{8} + 2 \times \frac{5}{8} \times \frac{1}{8}$, $P(S=5) = 2 \times \frac{1}{8} \times \frac{2}{8}$ | **M1** | 2nd M1 for $2pq$ for $P(S=3)$ or $P(S=5)$, or $p^2 + 2pq$ for $P(S=4)$ |
| Complete table: $P(S=2)=\frac{25}{64}$, $P(S=3)=\frac{20}{64}$, $P(S=4)=\frac{14}{64}$, $P(S=5)=\frac{4}{64}$, $P(S=6)=\frac{1}{64}$ | **A1** | 2nd A1 for complete correct probability distribution (allow awrt 3sf); Note: if 5 values of $S$ not found, can score B0M1A1M1A0 |
| | **(5)** | |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Let $X$ = number of scoops ordered by $n$ customers; $P(X=n) = \left(\frac{5}{8}\right)^n$ | | |
| $P(X > n) = 1 - \left(\frac{5}{8}\right)^n$ | **M1** | 1st M1 for $1 - p^n > 0.99$ (allow equation) or equivalent e.g. $p^n < 0.01$ |
| $1 - \left(\frac{5}{8}\right)^n > 0.99$ | | |
| $\left(\frac{5}{8}\right)^n < 0.01$ | **dM1** | 2nd dM1 dependent on 1st M1 for correct use of a valid method to solve to $n > k$ (allow equation) |
| $n > 9.798\ldots$ | | |
| $n = 10$ | **A1cao** | A1 for 10 cao |
| | **(3)** | |
| | **[8 marks]** | |\begin{enumerate}
\item An ice cream shop sells a large number of 1 scoop, 2 scoop and 3 scoop ice cream cones to its customers in the ratio $5 : 2 : 1$
\end{enumerate}
A random sample of 2 customers at the ice cream shop is taken. Each customer orders a 1 scoop or a 2 scoop or a 3 scoop ice cream cone.
Let $S$ represent the total number of ice cream scoops ordered by these 2 customers.\\
(a) Find the sampling distribution of $S$
A random sample of $n$ customers at the ice cream shop is taken. Each customer orders a 1 scoop or a 2 scoop or a 3 scoop ice cream cone. The probability that more than $n$ scoops of ice cream are ordered by these customers is greater than 0.99\\
(b) Find the smallest possible value of $n$\\
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\hfill \mbox{\textit{Edexcel S2 2016 Q7 [8]}}