| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Sampling distribution of range or maximum |
| Difficulty | Challenging +1.2 This question requires systematic enumeration of sample outcomes and solving simultaneous equations from probability conditions. While it involves multiple steps and careful algebraic manipulation, the conceptual framework (sampling with replacement, maximum of two values) is standard S2 material with no novel insights required. The arithmetic is somewhat involved but methodical. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| \(m\) | 1 | 5 | 10 |
| \(\mathrm { P } ( M = m )\) | \(\frac { 1 } { 25 }\) | \(\frac { 13 } { 80 }\) | \(\frac { 319 } { 400 }\) |
| Answer | Marks | Guidance |
|---|---|---|
| \((1,1)\), \((1,5)[x2]\), \((5,5)\), \((1,10)[x2]\), \((5,10)[x2]\), \((10,10)\) e.g. \((1,5)\) and \((5,1)\) counts once only | B2 | B2 all 6 pairs correct, ignore duplicates but no incorrect pairs. B1 at least 4 correct pairs. |
| Answer | Marks | Guidance |
|---|---|---|
| [For \(M=1\), \((1,1)\)] \(q \times q = \frac{1}{25}\), \(q = \frac{1}{5}\) | M1, A1 | |
| [For \(M=5\), \((1,5),(5,1),(5,5)\)] \(qr + rq + r^2 = \frac{13}{80}\) | M1 | |
| \(r^2 + 2\left(\frac{1}{5}\right)r - \frac{13}{80} = 0 \rightarrow r = \frac{-\frac{2}{5} + \sqrt{(\frac{2}{5})^2 - 4(-\frac{13}{80})}}{2} \rightarrow r = \frac{1}{4}\) | M1 A1 | \(3^{rd}\) M1 attempt to solve quadratic (formula, completing square or factorising) |
| [For \(M=10\)] \(2qs + 2rs + s^2 = \frac{319}{400}\) or \(q + r + s = 1\) | M1 | |
| \(s = \frac{11}{20}\) | A1 |
## Question 5:
### Part (a):
$(1,1)$, $(1,5)[x2]$, $(5,5)$, $(1,10)[x2]$, $(5,10)[x2]$, $(10,10)$ e.g. $(1,5)$ and $(5,1)$ counts once only | B2 | B2 all 6 pairs correct, ignore duplicates but no incorrect pairs. B1 at least 4 correct pairs.
### Part (b):
[For $M=1$, $(1,1)$] $q \times q = \frac{1}{25}$, $q = \frac{1}{5}$ | M1, A1 |
[For $M=5$, $(1,5),(5,1),(5,5)$] $qr + rq + r^2 = \frac{13}{80}$ | M1 |
$r^2 + 2\left(\frac{1}{5}\right)r - \frac{13}{80} = 0 \rightarrow r = \frac{-\frac{2}{5} + \sqrt{(\frac{2}{5})^2 - 4(-\frac{13}{80})}}{2} \rightarrow r = \frac{1}{4}$ | M1 A1 | $3^{rd}$ M1 attempt to solve quadratic (formula, completing square or factorising)
[For $M=10$] $2qs + 2rs + s^2 = \frac{319}{400}$ or $q + r + s = 1$ | M1 |
$s = \frac{11}{20}$ | A1 |
**SC:** B1 for $\frac{q}{q+r+s} = \frac{1}{5}$, B1 for $\frac{r}{q+r+s} = \frac{1}{4}$, B1 for $\frac{s}{q+r+s} = \frac{11}{20}$
**Solving 3TQ:** Formula: if correct formula quoted allow 1 slip. Complete Sq: $(r+\frac{1}{5})^2 - \frac{1}{25} - \frac{13}{50} = 0$, allow 1 slip. Factorise: must multiply out to give "ends [inc. sign]" or "middle term".
---5. A bag contains a large number of coins. It contains only $1 \mathrm { p } , 5 \mathrm { p }$ and 10 p coins. The fraction of 1 p coins in the bag is $q$, the fraction of 5 p coins in the bag is $r$ and the fraction of 10p coins in the bag is $s$.
Two coins are selected at random from the bag and the coin with the highest value is recorded. Let $M$ represent the value of the highest coin.
The sampling distribution of $M$ is given below
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
$m$ & 1 & 5 & 10 \\
\hline
$\mathrm { P } ( M = m )$ & $\frac { 1 } { 25 }$ & $\frac { 13 } { 80 }$ & $\frac { 319 } { 400 }$ \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item List all the possible samples of two coins which may be selected.
\item Find the value of $q$, the value of $r$ and the value of $s$
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2016 Q5 [9]}}