| Exam Board | OCR MEI |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Combinations & Selection |
| Type | Implementing simple random or systematic sampling |
| Difficulty | Moderate -0.8 This is a straightforward algorithm execution question requiring basic arithmetic and understanding of complementary probability. Part (i) involves simple repeated calculations with no conceptual difficulty. Parts (ii-iv) test interpretation and simple modification of the algorithm, which are routine D1 skills. The birthday problem context is well-known and the modification required is trivial (change 26 to 365). This is easier than average A-level maths due to minimal calculation complexity and no problem-solving insight required. |
| Spec | 7.03a Algorithm definition: input, output, deterministic, finite7.03b Algorithm awareness: uses and practical limitations7.03c Working with algorithms: trace, interpret, adapt |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(n\): 1, 2, 3, 4, 5, 6, 7 | M1 | \(n=2\ldots\) awrt 0.96 |
| \(p\): 1, 0.962, 0.888, 0.785, 0.664, 0.537, 0.413 | M1 | \(n=3\ldots\) awrt 0.88 or 0.89 |
| A1 | \(n=4\ldots\) awrt 0.79 | |
| A1 | stopping at \(n=7\) with \(p<0.5\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Need to select 7 cards for the probability of repetition on the list to exceed 0.5 | B1 | their "7" |
| B1 | P(repetition) exceeds 0.5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Step 1: Set \(n = 1\). Step 2: Set \(p = 1\). Step 3: Set \(n = n + 1\). Step 4: Set \(p = p \times (366-n)/365\). Step 5: If \(p < 0.5\) then stop. Step 6: Go to Step 3. | B1 | both changes (step 4) and no others |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Because they do not have the same frequency of occurrence (probability) as other birthdays. | B1 |
# Question 2:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $n$: 1, 2, 3, 4, 5, 6, 7 | M1 | $n=2\ldots$ awrt 0.96 |
| $p$: 1, 0.962, 0.888, 0.785, 0.664, 0.537, 0.413 | M1 | $n=3\ldots$ awrt 0.88 or 0.89 |
| | A1 | $n=4\ldots$ awrt 0.79 |
| | A1 | stopping at $n=7$ with $p<0.5$ |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Need to select 7 cards for the probability of repetition on the list to exceed 0.5 | B1 | their "7" |
| | B1 | P(repetition) exceeds 0.5 |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Step 1: Set $n = 1$. Step 2: Set $p = 1$. Step 3: Set $n = n + 1$. Step 4: Set $p = p \times (366-n)/365$. Step 5: If $p < 0.5$ then stop. Step 6: Go to Step 3. | B1 | both changes (step 4) and no others |
## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Because they do not have the same frequency of occurrence (probability) as other birthdays. | B1 | |
---2 A bag contains 26 cards. A different letter of the alphabet is written on each one. A card is chosen at random and its letter is written down. The card is returned to the bag. The bag is shaken and the process is repeated several times.
Tania wants to investigate the probability of a letter appearing twice. She wants to know how many cards need to be chosen for this probability to exceed 0.5. Tania uses the following algorithm.
Step 1 Set $n = 1$\\
Step 2 Set $p = 1$\\
Step 3 Set $n = n + 1$\\
Step 4 Set $p = p \times \left( \frac { 27 - n } { 26 } \right)$\\
Step 5 If $p < 0.5$ then stop\\
Step 6 Go to Step 3\\
(i) Run the algorithm.\\
(ii) Interpret your results.
A well-known problem asks how many randomly-chosen people need to be assembled in a room before the probability of at least two of them sharing a birthday exceeds 0.5 (ignoring anyone born on 29 February).\\
(iii) Modify Tania's algorithm to answer the birthday problem. (Do not attempt to run your modified algorithm.)\\
(iv) Why have 29 February birthdays been excluded?
\hfill \mbox{\textit{OCR MEI D1 2016 Q2 [8]}}