| Exam Board | OCR MEI |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modelling and Hypothesis Testing |
| Type | Multi-stage probability simulation |
| Difficulty | Moderate -0.8 This is a straightforward simulation exercise requiring only basic probability understanding (p=1/3) and following mechanical instructions with random number tables. The multi-part structure adds length but not conceptual difficulty—each step is routine application of the same simple rule with no problem-solving, proof, or mathematical insight required. |
| Answer | Marks | Guidance |
|---|---|---|
| (i) e.g. 0, 1, 2 → fall; 3, 4, 5, 6, 7, 8 → not fall; 9 → redraw | M1 | ignore at least 1 proportions |
| A1 | correct | |
| A1 | efficient | |
| (ii) | apple | r n |
| 1 | 1 | yes |
| 2 | 3 | no |
| 3 | 8 | no |
| 4 | 0 | yes |
| 5 | 2 | yes |
| 6 | 7 | no |
| Three apples fall in this simulation. | M1 A2 | −1 each error |
| B1∇ | ||
| (iii) | apple | r n |
| 2 | 0 | yes |
| 3 | 1 | yes |
| 6 | 4 | no |
| apple | r n | fall? |
| 6 | 4 | no |
| apple | r n | fall? |
| 6 | 8 | no |
| apple | r n | fall? |
| 6 | 0 | yes |
| M1 A2 | −1 each error | |
| A1∇ | ||
| (iv) | apple | r n |
| 1 | picked | |
| 2 | 1 | yes |
| 3 | 3 | no |
| 4 | 8 | no |
| 5 | 0 | yes |
| 6 | 2 | yes |
| apple | r n | fall? |
| 3 | picked | |
| 4 | 7 | no |
| apple | r n | fall? |
| 4 | picked | |
| M1 A2 | −1 each error | |
| B1∇ | ||
| (v) more simulations | B1 |
**(i)** e.g. 0, 1, 2 → fall; 3, 4, 5, 6, 7, 8 → not fall; 9 → redraw | M1 | ignore at least 1 proportions
| A1 | correct
| A1 | efficient
**(ii)** | apple | r n | fall? |
| 1 | 1 | yes |
| 2 | 3 | no |
| 3 | 8 | no |
| 4 | 0 | yes |
| 5 | 2 | yes |
| 6 | 7 | no |
Three apples fall in this simulation. | M1 A2 | −1 each error
| B1∇ |
**(iii)** | apple | r n | fall? |
| 2 | 0 | yes |
| 3 | 1 | yes |
| 6 | 4 | no |
| apple | r n | fall? |
| 6 | 4 | no |
| apple | r n | fall? |
| 6 | 8 | no |
| apple | r n | fall? |
| 6 | 0 | yes | 5 days before all have fallen |
| M1 A2 | −1 each error
| A1∇ |
**(iv)** | apple | r n | fall? |
| 1 | | picked |
| 2 | 1 | yes |
| 3 | 3 | no |
| 4 | 8 | no |
| 5 | 0 | yes |
| 6 | 2 | yes |
| apple | r n | fall? |
| 3 | | picked |
| 4 | 7 | no |
| apple | r n | fall? |
| 4 | | picked | 3 days before none left |
| M1 A2 | −1 each error
| B1∇ |
**(v)** more simulations | B1 |6 An apple tree has 6 apples left on it. Each day each remaining apple has a probability of $\frac { 1 } { 3 }$ of falling off the tree during the day.\\
(i) Give a rule for using one-digit random numbers to simulate whether or not a particular apple falls off the tree during a given day.\\
(ii) Use the random digits given in your answer book to simulate how many apples fall off the tree during day 1 . Give the total number of apples that fall during day 1 .\\
(iii) Continue your simulation from the end of day 1 , which you simulated in part (ii), for successive days until there are no apples left on the tree. Use the same list of random digits, continuing from where you left off in part (ii).
During which day does the last apple fall from the tree?
Now suppose that at the start of each day the gardener picks one apple from the tree and eats it.\\
(iv) Repeat your simulation with the gardener picking the lowest numbered apple remaining on the tree at the start of each day. Give the day during which the last apple falls or is picked. Use the same string of random digits, a copy of which is provided for your use in this part of the question.\\
(v) How could your results be made more reliable?
\hfill \mbox{\textit{OCR MEI D1 2010 Q6 [16]}}