| Exam Board | OCR MEI |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modelling and Hypothesis Testing |
| Type | Multi-stage probability simulation |
| Difficulty | Moderate -0.5 This is a straightforward simulation exercise requiring students to apply given probability rules to random numbers and count outcomes. While it involves multiple steps and careful bookkeeping, it requires no mathematical insight or problem-solving—just methodical application of simple comparison rules (e.g., '00-09 means failure'). The conceptual demand is low for A-level, making it easier than average. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
| Year of life | first | second | third | fourth | fifth | sixth | ||
| 0.10 | 0.05 | 0.02 | 0.20 | 0.20 | 0.30 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Year 1: 00–09 failure, otherwise no failure | M1 A1 | |
| Year 2: 00–04 | A1 | |
| Year 3: 00–01 | ||
| Year 4: 00–19 | ||
| Year 5: 00–19 | ||
| Year 6: 00–29 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Completed table with ticks and crosses as shown | M1 A1 A1 A1 B1 | ticks and crosses; run 1; runs 2–4; runs 5–7; runs 8–10 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| 0.6 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| If no failure then continue after year 3 – but using rules for yrs 1 to 3 | B1 B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Completed table with ticks and crosses as shown | M1 A1 A1 | runs 1–5; runs 6–10 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| 0.3 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| More repetitions | B1 |
**Part (i)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Year 1: 00–09 failure, otherwise no failure | M1 A1 | |
| Year 2: 00–04 | A1 | |
| Year 3: 00–01 | | |
| Year 4: 00–19 | | |
| Year 5: 00–19 | | |
| Year 6: 00–29 | | |
**Part (ii)(A)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Completed table with ticks and crosses as shown | M1 A1 A1 A1 B1 | ticks and crosses; run 1; runs 2–4; runs 5–7; runs 8–10 |
**Part (ii)(B)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| 0.6 | B1 | |
**Part (iii)(A)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| If no failure then continue after year 3 – but using rules for yrs 1 to 3 | B1 B1 | |
**Part (iii)(B)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Completed table with ticks and crosses as shown | M1 A1 A1 | runs 1–5; runs 6–10 |
**Part (iii)(C)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| 0.3 | B1 | |
**Part (iv)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| More repetitions | B1 | |6 Answer parts (ii)(A) and (iii)(B) of this question on the insert provided.
A particular component of a machine sometimes fails. The probability of failure depends on the age of the component, as shown in Table 6.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
Year of life & first & second & third & fourth & fifth & sixth \\
\hline
\begin{tabular}{ l }
Probability of failure during year, \\
given no earlier failure \\
\end{tabular} & 0.10 & 0.05 & 0.02 & 0.20 & 0.20 & 0.30 \\
\hline
\end{tabular}
\end{center}
\section*{Table 6}
You are to simulate six years of machine operation to estimate the probability of the component failing during that time. This will involve you using six 2-digit random numbers, one for each year.
\begin{enumerate}[label=(\roman*)]
\item Give a rule for using a 2-digit random number to simulate failure of the component in its first year of life.
Similarly give rules for simulating failure during each of years 2 to 6 .
\item (A) Use your rules, together with the random numbers given in the insert, to complete the simulation table in the insert. This simulates 10 repetitions of six years operation of the machine. Start in the first column working down cell-by-cell. In each cell enter a tick if there is no simulated failure and a cross if there is a simulated failure.
Stop and move on to the next column if a failure occurs.\\
(B) Use your results to estimate the probability of a failure occurring.
It is suggested that any component that has not failed during the first three years of its life should automatically be replaced.
\item (A) Describe how to simulate the operation of this policy.\\
(B) Use the table in the insert to simulate 10 repetitions of the application of this policy. Re-use the same random numbers that are given in the insert.\\
(C) Use your results to estimate the probability of a failure occurring.
\item How might the reliability of your estimates in parts (ii) and (iii) be improved?
\end{enumerate}
\hfill \mbox{\textit{OCR MEI D1 2006 Q6 [16]}}