| Exam Board | OCR MEI |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2015 |
| 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 application of Markov chain simulation using random number tables. Parts (i) and (iii) require simple assignment of digit ranges to probabilities (routine procedure), while parts (ii) and (iv) involve mechanical execution of the simulation rules. No conceptual depth, proof, or problem-solving insight is required—just following a standard algorithm taught in D1. |
| Spec | 2.01c Sampling techniques: simple random, opportunity, etc2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space |
| \backslashbox{last meal out}{next meal out} | French | Greek | Hungarian |
| French | \(\frac { 1 } { 5 }\) | \(\frac { 3 } { 5 }\) | \(\frac { 1 } { 5 }\) |
| Greek | \(\frac { 1 } { 2 }\) | \(\frac { 3 } { 10 }\) | \(\frac { 1 } { 5 }\) |
| Hungarian | \(\frac { 1 } { 3 }\) | \(\frac { 1 } { 3 }\) | \(\frac { 1 } { 3 }\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| French: \(0,1,2,3,4,5,6 \rightarrow\) Greek; \(7,8,9 \rightarrow\) French | B1 | French |
| Greek: \(0,1,2,3,4,5 \rightarrow\) French; \(6,7,8,9 \rightarrow\) Greek | M1 | proportions |
| A1 | efficient | |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Using Greek rule | M1 | Greek |
| Using French rule | M1 | French |
| e.g. F G G G F G F G G G | A1\(\checkmark\) | |
| Computing observed probabilities | ||
| e.g. \(P(F)=0.3\) and \(P(G)=0.7\) (Long run probabilities are \(\frac{6}{13}\) French and \(\frac{7}{13}\) Greek) | B1\(\checkmark\) | |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| French: \(0,1 \rightarrow\) French; \(2,3,4,5,6,7 \rightarrow\) Greek; \(8,9 \rightarrow\) Hungarian | B1 | |
| Greek: \(0,1,2,3,4 \rightarrow\) French; \(5,6,7 \rightarrow\) Greek; \(8,9 \rightarrow\) Hungarian | B1 | |
| Hungarian: \(0,1,2 \rightarrow\) French; \(3,4,5 \rightarrow\) Greek; \(6,7,8 \rightarrow\) Hungarian; \(9 \rightarrow\) reject and redraw | M1 | reject one (or more) |
| A1 | proportions | |
| A1 | efficient | |
| [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Greek rule applied in correct circumstances and correctly | B1 | |
| French rule applied in correct circumstances and correctly | B1 | |
| Hungarian rule applied in correct circumstances and correctly | B1 | |
| e.g. F F H F G H F G F F; so \(P(F)=0.6\), \(P(G)=0.2\), \(P(H)=0.2\) (Long run proportions are \(\frac{56}{169}\), \(\frac{74}{169}\) and \(\frac{39}{169}\)) | B1\(\checkmark\) | |
| [4] |
# Question 6:
## Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| French: $0,1,2,3,4,5,6 \rightarrow$ Greek; $7,8,9 \rightarrow$ French | B1 | French |
| Greek: $0,1,2,3,4,5 \rightarrow$ French; $6,7,8,9 \rightarrow$ Greek | M1 | proportions |
| | A1 | efficient |
| | **[3]** | |
## Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Using Greek rule | M1 | Greek |
| Using French rule | M1 | French |
| e.g. F G G G F G F G G G | A1$\checkmark$ | |
| Computing observed probabilities | | |
| e.g. $P(F)=0.3$ and $P(G)=0.7$ (Long run probabilities are $\frac{6}{13}$ French and $\frac{7}{13}$ Greek) | B1$\checkmark$ | |
| | **[4]** | |
## Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| French: $0,1 \rightarrow$ French; $2,3,4,5,6,7 \rightarrow$ Greek; $8,9 \rightarrow$ Hungarian | B1 | |
| Greek: $0,1,2,3,4 \rightarrow$ French; $5,6,7 \rightarrow$ Greek; $8,9 \rightarrow$ Hungarian | B1 | |
| Hungarian: $0,1,2 \rightarrow$ French; $3,4,5 \rightarrow$ Greek; $6,7,8 \rightarrow$ Hungarian; $9 \rightarrow$ reject and redraw | M1 | reject one (or more) |
| | A1 | proportions |
| | A1 | efficient |
| | **[5]** | |
## Part (iv):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Greek rule applied in correct circumstances and correctly | B1 | |
| French rule applied in correct circumstances and correctly | B1 | |
| Hungarian rule applied in correct circumstances and correctly | B1 | |
| e.g. F F H F G H F G F F; so $P(F)=0.6$, $P(G)=0.2$, $P(H)=0.2$ (Long run proportions are $\frac{56}{169}$, $\frac{74}{169}$ and $\frac{39}{169}$) | B1$\checkmark$ | |
| | **[4]** | |6 Adrian and Kleo like to go out for meals, sometimes to a French restaurant, and sometimes to a Greek restaurant. If their last meal out was at the French restaurant, then the probability of their next meal out being at the Greek restaurant is 0.7 , whilst the probability of it being at the French restaurant is 0.3 . If their last meal out was at the Greek restaurant, then the probability of their next meal out being at the French restaurant is 0.6 , whilst the probability of it being at the Greek restaurant is 0.4 .\\
(i) Construct two simulation rules, each using single-digit random numbers, to model their choices of where to eat.\\
(ii) Their last meal out was at the Greek restaurant. Use the random digits printed in your answer book to simulate their choices for the next 10 of their meals out. Hence estimate the proportion of their meals out which are at the French restaurant, and the proportion which are at the Greek restaurant.
Adrian and Kleo find a Hungarian restaurant which they like. The probabilities of where they eat next are now given in the following table.
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
\backslashbox{last meal out}{next meal out} & French & Greek & Hungarian \\
\hline
French & $\frac { 1 } { 5 }$ & $\frac { 3 } { 5 }$ & $\frac { 1 } { 5 }$ \\
\hline
Greek & $\frac { 1 } { 2 }$ & $\frac { 3 } { 10 }$ & $\frac { 1 } { 5 }$ \\
\hline
Hungarian & $\frac { 1 } { 3 }$ & $\frac { 1 } { 3 }$ & $\frac { 1 } { 3 }$ \\
\hline
\end{tabular}
\end{center}
(iii) Construct simulation rules, each using single-digit random numbers, to model this new situation.\\
(iv) Their last meal out was at the Greek restaurant. Use the random digits printed in your answer book to simulate their choices for the next 10 of their meals out. Hence estimate the proportion of their meals out which are at each restaurant.
\section*{END OF QUESTION PAPER}
\hfill \mbox{\textit{OCR MEI D1 2015 Q6 [16]}}