| Exam Board | OCR MEI |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2008 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modelling and Hypothesis Testing |
| Type | Queueing and service simulation |
| Difficulty | Moderate -0.8 This is a standard D1 simulation question requiring straightforward application of random number allocation rules and table completion. While it involves multiple steps, each step follows routine procedures (assigning random number ranges to probabilities, running simulations) with no conceptual difficulty or novel problem-solving required. The calculations are arithmetic and the methodology is textbook-standard for this module. |
| Spec | 2.03a Mutually exclusive and independent events7.03c Working with algorithms: trace, interpret, adapt |
| Time interval between cars (minutes) | 1 | 2 | 3 | 4 | 5 |
| Probability | \(\frac { 1 } { 10 }\) | \(\frac { 3 } { 10 }\) | \(\frac { 2 } { 5 }\) | \(\frac { 1 } { 10 }\) | \(\frac { 1 } { 10 }\) |
| Number of passengers per car | 1 | 2 | 3 | 4 | 5 | 6 |
| Probability | \(\frac { 1 } { 6 }\) | \(\frac { 1 } { 3 }\) | \(\frac { 1 } { 12 }\) | \(\frac { 1 } { 4 }\) | \(\frac { 1 } { 12 }\) | \(\frac { 1 } { 12 }\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(00–09 \to +1\); \(10–39 \to +2\); \(40–79 \to +3\); \(80–89 \to +4\); \(90–99 \to +5\) | M1 A1 A1 | proportions OK; efficient |
| Answer | Marks | Guidance |
|---|---|---|
| \(00–15 \to +1\); \(16–47 \to +2\); \(48–55 \to +3\); \(56–79 \to +4\); \(80–87 \to +5\); \(88–95 \to +6\); \(96, 97, 98, 99\) reject | M1 A2 A1 | some rejected; proportions OK (\(-1\) each error); efficient |
| Answer | Marks | Guidance |
|---|---|---|
| Simulation table with 10 runs showing cars arriving after Joe, time intervals, number of passengers, and time to 15 passengers. Entries: Run 1-10 with various combinations showing final time to wait column. | M1 A2 M1 A1 A1 A1 | (\(-1\) each error); simulation; time intervals; passengers; time to wait |
| Answer | Marks |
|---|---|
| 0.8; more runs | B1 B1 |
**(i)**
| $00–09 \to +1$; $10–39 \to +2$; $40–79 \to +3$; $80–89 \to +4$; $90–99 \to +5$ | M1 A1 A1 | proportions OK; efficient |
**(ii)**
| $00–15 \to +1$; $16–47 \to +2$; $48–55 \to +3$; $56–79 \to +4$; $80–87 \to +5$; $88–95 \to +6$; $96, 97, 98, 99$ reject | M1 A2 A1 | some rejected; proportions OK ($-1$ each error); efficient |
**(iii) & (iv)**
| Simulation table with 10 runs showing cars arriving after Joe, time intervals, number of passengers, and time to 15 passengers. Entries: Run 1-10 with various combinations showing final time to wait column. | M1 A2 M1 A1 A1 A1 | ($-1$ each error); simulation; time intervals; passengers; time to wait |
**(v)**
| 0.8; more runs | B1 B1 | |4 Joe is to catch a plane to go on holiday. He has arranged to leave his car at a car park near to the airport. There is a bus service from the car park to the airport, and the bus leaves when there are at least 15 passengers on board. Joe is delayed getting to the car park and arrives needing the bus to leave within 15 minutes if he is to catch his plane. He is the $10 ^ { \text {th } }$ passenger to board the bus, so he has to wait for another 5 passengers to arrive.
The distribution of the time intervals between car arrivals and the distribution of the number of passengers per car are given below.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
Time interval between cars (minutes) & 1 & 2 & 3 & 4 & 5 \\
\hline
Probability & $\frac { 1 } { 10 }$ & $\frac { 3 } { 10 }$ & $\frac { 2 } { 5 }$ & $\frac { 1 } { 10 }$ & $\frac { 1 } { 10 }$ \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
Number of passengers per car & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
Probability & $\frac { 1 } { 6 }$ & $\frac { 1 } { 3 }$ & $\frac { 1 } { 12 }$ & $\frac { 1 } { 4 }$ & $\frac { 1 } { 12 }$ & $\frac { 1 } { 12 }$ \\
\hline
\end{tabular}
\end{center}
(i) Give an efficient rule for using 2-digit random numbers to simulate the intervals between car arrivals.\\
(ii) Give an efficient rule for using 2-digit random numbers to simulate the number of passengers in a car.\\
(iii) The incomplete table in your answer book shows the results of nine simulations of the situation. Complete the table, showing in each case whether or not Joe catches his plane.\\
(iv) Use the random numbers provided in your answer book to run a tenth simulation.\\[0pt]
(v) Estimate the probability of Joe catching his plane. State how you could improve your estimate. [2]
\hfill \mbox{\textit{OCR MEI D1 2008 Q4 [16]}}