| Exam Board | OCR MEI |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modelling and Hypothesis Testing |
| Type | Markov chain transition simulation |
| Difficulty | Moderate -0.3 This is a straightforward Markov chain simulation question requiring students to assign random number ranges to probabilities and execute a simple simulation. While it involves multiple parts, each step is routine: converting fractions to cumulative ranges (standard D1 technique), following a deterministic simulation with given random numbers, and making basic criticisms of models. The mathematical content is procedural rather than requiring problem-solving or insight, making it slightly easier than average A-level. |
| Spec | 2.03c Conditional probability: using diagrams/tables7.03c Working with algorithms: trace, interpret, adapt |
| \cline { 3 - 5 } \multicolumn{2}{c|}{} | next day's weather | ||||||
| \cline { 3 - 5 } \multicolumn{2}{c|}{} | dry | wet | snowy | ||||
\multirow{3}{*}{
| dry | \(\frac { 4 } { 10 }\) | \(\frac { 3 } { 10 }\) | \(\frac { 3 } { 10 }\) | |||
| \cline { 2 - 5 } | wet | \(\frac { 2 } { 10 }\) | \(\frac { 5 } { 10 }\) | \(\frac { 3 } { 10 }\) | |||
| \cline { 2 - 5 } | snowy | \(\frac { 2 } { 7 }\) | \(\frac { 2 } { 7 }\) | \(\frac { 3 } { 7 }\) | |||
| Answer | Marks | Guidance |
|---|---|---|
| Dry: 00–39; Wet: 40–69; Snowy: 70–99 | B2 | B1 for correct ranges for at least 2 states |
| Answer | Marks | Guidance |
|---|---|---|
| Dry: 00–19; Wet: 20–69; Snowy: 70–99 | B2 | B1 for correct ranges for at least 2 states |
| Answer | Marks | Guidance |
|---|---|---|
| Dry: 00–27; Wet: 28–55; Snowy: 56–99 (using \(\frac{2}{7}, \frac{2}{7}, \frac{3}{7}\)) | B3 | B1 for correct ranges for at least 2 states; must be efficient (no wasted numbers) |
| Answer | Marks |
|---|---|
| M1 for correct use of rules | A3 for all 7 days correct (allow follow-through from part (i)) |
| Answer | Marks |
|---|---|
| From simulation: 2 dry days out of 7 (days 0 and 1, or just days 1–7 depending on interpretation); proportion \(= \frac{2}{7}\) (or \(\frac{1}{7}\)) | B1 follow-through from part (ii) |
| Answer | Marks | Guidance |
|---|---|---|
| Use more random numbers / simulate over a longer period / repeat simulation many times and average the results | B2 | B1 per valid point |
| Answer | Marks | Guidance |
|---|---|---|
| - Probabilities estimated from limited data and may not be accurate | B2 | B1 per valid criticism |
# Question 6:
## Part (i)(A) – today dry
Dry: 00–39; Wet: 40–69; Snowy: 70–99 | B2 | B1 for correct ranges for at least 2 states
## Part (i)(B) – today wet
Dry: 00–19; Wet: 20–69; Snowy: 70–99 | B2 | B1 for correct ranges for at least 2 states
## Part (i)(C) – today snowy
Dry: 00–27; Wet: 28–55; Snowy: 56–99 (using $\frac{2}{7}, \frac{2}{7}, \frac{3}{7}$) | B3 | B1 for correct ranges for at least 2 states; must be efficient (no wasted numbers)
## Part (ii)
Starting dry, using random numbers 23, 85, 98, 99, 56, 47, 82, 14, 03, 12 (need 7 days):
- Day 0: Dry
- 23 → Dry (00–39): Day 1 Dry
- 85 → Snowy (70–99): Day 2 Snowy
- 98 → Snowy (56–99): Day 3 Snowy
- 99 → Snowy (56–99): Day 4 Snowy
- 56 → Snowy (56–99): Day 5 Snowy
- 47 → Wet (28–55): Day 6 Wet
- 82 → Snowy (70–99): Day 7 Snowy
| M1 for correct use of rules | A3 for all 7 days correct (allow follow-through from part (i))
## Part (iii)
From simulation: 2 dry days out of 7 (days 0 and 1, or just days 1–7 depending on interpretation); proportion $= \frac{2}{7}$ (or $\frac{1}{7}$) | B1 follow-through from part (ii)
## Part (iv)
Use more random numbers / simulate over a longer period / repeat simulation many times and average the results | B2 | B1 per valid point
## Part (v)
Any two valid criticisms, e.g.:
- Only three weather states — weather is more varied/continuous
- Probabilities may not be constant throughout winter / may vary by month or year
- Next day's weather may depend on more than just the current day (not truly Markov)
- Probabilities estimated from limited data and may not be accurate | B2 | B1 per valid criticism6 In winter in Metland the weather each day can be classified as dry, wet or snowy. The table shows the probabilities for the next day's weather given the current day's weather.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | }
\cline { 3 - 5 }
\multicolumn{2}{c|}{} & \multicolumn{3}{|c|}{next day's weather} \\
\cline { 3 - 5 }
\multicolumn{2}{c|}{} & dry & wet & snowy \\
\hline
\multirow{3}{*}{\begin{tabular}{ l }
current \\
day's \\
weather \\
\end{tabular}} & dry & $\frac { 4 } { 10 }$ & $\frac { 3 } { 10 }$ & $\frac { 3 } { 10 }$ \\
\cline { 2 - 5 }
& wet & $\frac { 2 } { 10 }$ & $\frac { 5 } { 10 }$ & $\frac { 3 } { 10 }$ \\
\cline { 2 - 5 }
& snowy & $\frac { 2 } { 7 }$ & $\frac { 2 } { 7 }$ & $\frac { 3 } { 7 }$ \\
\hline
\end{tabular}
\end{center}
You are to use two-digit random numbers to simulate the winter weather in Metland.
\begin{enumerate}[label=(\roman*)]
\item Give an efficient rule for using two-digit random numbers to simulate tomorrow's weather if today is\\
(A) dry,\\
(B) wet,\\
(C) snowy.
\item Today is a dry winter's day in Metland. Use the following two-digit random numbers to simulate the next 7 days' weather in Metland.
$$\begin{array} { l l l l l l l l l l }
23 & 85 & 98 & 99 & 56 & 47 & 82 & 14 & 03 & 12
\end{array}$$
\item Use your simulation from part (ii) to estimate the proportion of dry days in a Metland winter.
\item Explain how you could use simulation to produce an improved estimate of the proportion of dry days in a Metland winter.
\item Give two criticisms of this model of weather.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI D1 2007 Q6 [16]}}