| Exam Board | OCR MEI |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2009 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modelling and Hypothesis Testing |
| Type | Markov chain transition simulation |
| Difficulty | Moderate -0.8 This is a straightforward simulation exercise requiring only basic understanding of random number assignment and recording outcomes. Students follow mechanical procedures (assign digits to outcomes, trace paths, count results) with no mathematical analysis, proof, or problem-solving insight required. The Markov chain context is superficial—students simply follow given probabilities through a diagram. |
| Spec | 2.03c Conditional probability: using diagrams/tables7.03c Working with algorithms: trace, interpret, adapt |
| Answer | Marks |
|---|---|
| (i) e.g. 0–4 exit, 5–9 other vertex | B1, B1 |
| (ii) Process with exits (see table for detailed working with rows 1–10 showing A, B, ExA, ExB combinations). Theoretical answers: \(\frac{2}{3}, \frac{1}{3}, 2\) (Gambler's ruin) | M1 process with exits, A1 probabilities, M1 duration |
| (iii) e.g. 0–2 exit, 3–5 next vertex in cycle, 6–8 other vertex, 9–ignore and re-draw | M1 ignore conditionality, DM1 equal prob, A1 efficient |
| (iv) Process with exits (see table for detailed working with rows 1–10 showing A, B, C, ExA, ExB, ExC combinations). Theoretical probs are \(0.5, 0.25, 0.25\) (Markov chain) | M1, A2 |
**(i)** e.g. 0–4 exit, 5–9 other vertex | B1, B1 |
**(ii)** Process with exits (see table for detailed working with rows 1–10 showing A, B, ExA, ExB combinations). Theoretical answers: $\frac{2}{3}, \frac{1}{3}, 2$ (Gambler's ruin) | M1 process with exits, A1 probabilities, M1 duration |
**(iii)** e.g. 0–2 exit, 3–5 next vertex in cycle, 6–8 other vertex, 9–ignore and re-draw | M1 ignore conditionality, DM1 equal prob, A1 efficient |
**(iv)** Process with exits (see table for detailed working with rows 1–10 showing A, B, C, ExA, ExB, ExC combinations). Theoretical probs are $0.5, 0.25, 0.25$ (Markov chain) | M1, A2 |4 The diagram represents a very simple maze with two vertices, A and B. At each vertex a rat either exits the maze or runs to the other vertex, each with probability 0.5 . The rat starts at vertex A .\\
\includegraphics[max width=\textwidth, alt={}, center]{dab87ac5-eda4-433f-b07a-0a609aca2f65-4_79_930_534_571}\\
(i) Describe how to use 1-digit random numbers to simulate this situation.\\
(ii) Use the random digits provided in your answer book to run 10 simulations, each starting at vertex A. Hence estimate the probability of the rat exiting at each vertex, and calculate the mean number of times it runs between vertices before exiting.
The second diagram represents a maze with three vertices, A, B and C. At each vertex there are three possibilities, and the rat chooses one, each with probability $1 / 3$. The rat starts at vertex A.\\
\includegraphics[max width=\textwidth, alt={}, center]{dab87ac5-eda4-433f-b07a-0a609aca2f65-4_566_889_1082_589}\\
(iii) Describe how to use 1-digit random numbers to simulate this situation.\\
(iv) Use the random digits provided in your answer book to run 10 simulations, each starting at vertex A. Hence estimate the probability of the rat exiting at each vertex.
\hfill \mbox{\textit{OCR MEI D1 2009 Q4 [16]}}