| Exam Board | OCR MEI |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | Sorting and searching algorithms |
| Difficulty | Moderate -0.8 This is a straightforward D1 question testing basic understanding of using dice to simulate probability distributions. Parts (i) and (ii) require simple mapping of die outcomes to match given probabilities (e.g., faces 1-3 → X=1, face 4 → X=2, faces 5-6 → X=3). Part (iii) tests recognition that John's method produces a sum (0-100) rather than a uniform 2-digit number (10-99), requiring only basic critique of methodology rather than complex problem-solving. |
| Spec | 2.04a Discrete probability distributions5.01a Permutations and combinations: evaluate probabilities |
| \(x\) | 1 | 2 | 3 |
| \(\operatorname { Probability } ( X = x )\) | \(\frac { 1 } { 2 }\) | \(\frac { 1 } { 6 }\) | \(\frac { 1 } { 3 }\) |
| \(y\) | 1 | 2 | 3 |
| \(\operatorname { Probability } ( Y = y )\) | \(\frac { 1 } { 2 }\) | \(\frac { 1 } { 4 }\) | \(\frac { 1 } { 4 }\) |
| Answer | Marks | Guidance |
|---|---|---|
| Part | Answer/Working | Marks |
| (i) | Example: \(1, 2, 3 \to 1\); \(4 \to 2\); \(5, 6 \to 3\) | M1, A1, A1 |
| (ii) | Example: \(1, 2 \to 1\); \(3 \to 2\); \(4 \to 3\); \((5, 6 \to \text{reject and throw again})\) | M1, A1, A1 |
| (iii) | Non uniform allows 100 | B1, B1 |
| **Part** | **Answer/Working** | **Marks** | **Guidance** |
|----------|-------------------|-----------|------------|
| (i) | Example: $1, 2, 3 \to 1$; $4 \to 2$; $5, 6 \to 3$ | M1, A1, A1 | Function with domain $\{1,2,3,4,5,6\}$ and range $\{1,2,3\}$ (special cases possible if correct); Proportions 3:2:1; All OK |
| (ii) | Example: $1, 2 \to 1$; $3 \to 2$; $4 \to 3$; $(5, 6 \to \text{reject and throw again})$ | M1, A1, A1 | Special cases possible if correct! e.g. allow throwing die twice and allocating correct proportions of 36. |
| (iii) | Non uniform allows 100 | B1, B1 | "101 values" OK; No credit for e.g. "3 is not a two-digit number" |
---3 John has a standard die in his pocket (ie a cube with its six faces labelled from 1 to 6).\\
(i) Describe how John can use the die to obtain realisations of the random variable $X$, defined below.
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
$x$ & 1 & 2 & 3 \\
\hline
$\operatorname { Probability } ( X = x )$ & $\frac { 1 } { 2 }$ & $\frac { 1 } { 6 }$ & $\frac { 1 } { 3 }$ \\
\hline
\end{tabular}
\end{center}
(ii) Describe how John can use the die to obtain realisations of the random variable $Y$, defined below.
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
$y$ & 1 & 2 & 3 \\
\hline
$\operatorname { Probability } ( Y = y )$ & $\frac { 1 } { 2 }$ & $\frac { 1 } { 4 }$ & $\frac { 1 } { 4 }$ \\
\hline
\end{tabular}
\end{center}
(iii) John attempts to use the die to obtain a realisation of a uniformly distributed 2-digit random number. He throws the die 20 times. Each time he records one less than the number showing. He then adds together his 20 recorded numbers.
Criticise John's methodology.
\hfill \mbox{\textit{OCR MEI D1 2011 Q3 [8]}}