| Exam Board | OCR MEI |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Probability Definitions |
| Type | Multiple independent trials |
| Difficulty | Moderate -0.8 This is a straightforward simulation question requiring basic understanding of random number assignment and following a prescribed doubling strategy. The mathematical demands are minimal (simple arithmetic, calculating a mean), and the question provides extensive scaffolding with clear step-by-step instructions. While it involves multiple parts, each part is routine and requires only recall of standard simulation techniques rather than problem-solving or insight. |
| Spec | 2.01c Sampling techniques: simple random, opportunity, etc2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| e.g. \(0, 1, 2, 3 \rightarrow\) win; \(4, 5, 6, 7, 8, 9 \rightarrow\) lose | M1 | correct rule |
| A1 | efficient rule; disallow \(1, 8, 3, 5 \rightarrow\) win; disallow \(6, 7, 1, 3 \rightarrow\) win | |
| or \(0, 1, 2, 3, 4, 5 \rightarrow\) lose; \(6, 7, 8, 9 \rightarrow\) win |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| In the worst case Pierre will suffer 3 consecutive losses, of £100, £200 and £400 respectively. He will then be unable to fund the next bet of £800. | B1 | 100, 200, 400 may be implied, e.g. by 700 lost or 300 left |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| e.g. \(1, 6, 4 \rightarrow\) win; \(8, 6, 4 \rightarrow\) lose; \(8, 7, 4 \rightarrow\) lose; \(3, 1, 1 \rightarrow\) win; \(5, 3, 2 \rightarrow\) win | M1 | correct identification of first win and of first loss |
| \(A1\sqrt{}\) | rest | |
| or \(1, 6, 4 \rightarrow\) win; \(8, 6, 4 \rightarrow\) win; \(8, 7, 4 \rightarrow\) win; \(3, 1, 1 \rightarrow\) lose; \(5, 3, 2 \rightarrow\) lose | M1 | numbers \(\rightarrow\) L/W for 1 6 4 and 8 6 4 |
| \(A1\sqrt{}\) | rest + interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(100 \times\) no of wins \(+ 700 \times\) no of losses | M1 | weighted sum or 5 monetary outcomes – implied OK (Their "700" OK here) |
| e.g. \(-\)£1100 ... e.g. £3900 | \(A1\sqrt{}\) | correct sum following their simulation but not their 700 |
| e.g. on average a loss of £220 per application of the strategy ... e.g. on average £780 left | \(A1\sqrt{}\) | their sum/5 |
# Question 1:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| e.g. $0, 1, 2, 3 \rightarrow$ win; $4, 5, 6, 7, 8, 9 \rightarrow$ lose | M1 | correct rule |
| | A1 | efficient rule; disallow $1, 8, 3, 5 \rightarrow$ win; disallow $6, 7, 1, 3 \rightarrow$ win |
| or $0, 1, 2, 3, 4, 5 \rightarrow$ lose; $6, 7, 8, 9 \rightarrow$ win | | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| In the worst case Pierre will suffer 3 consecutive losses, of £100, £200 and £400 respectively. He will then be unable to fund the next bet of £800. | B1 | 100, 200, 400 may be implied, e.g. by 700 lost or 300 left |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| e.g. $1, 6, 4 \rightarrow$ win; $8, 6, 4 \rightarrow$ lose; $8, 7, 4 \rightarrow$ lose; $3, 1, 1 \rightarrow$ win; $5, 3, 2 \rightarrow$ win | M1 | correct identification of first win and of first loss |
| | $A1\sqrt{}$ | rest |
| or $1, 6, 4 \rightarrow$ win; $8, 6, 4 \rightarrow$ win; $8, 7, 4 \rightarrow$ win; $3, 1, 1 \rightarrow$ lose; $5, 3, 2 \rightarrow$ lose | M1 | numbers $\rightarrow$ L/W for 1 6 4 and 8 6 4 |
| | $A1\sqrt{}$ | rest + interpretation |
## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $100 \times$ no of wins $+ 700 \times$ no of losses | M1 | weighted sum or 5 monetary outcomes – implied OK (Their "700" OK here) |
| e.g. $-$£1100 ... e.g. £3900 | $A1\sqrt{}$ | correct sum following their simulation but not their 700 |
| e.g. on average a loss of £220 per application of the strategy ... e.g. on average £780 left | $A1\sqrt{}$ | their sum/5 |
---1 Pierre knows that, if he gambles, he will lose money in the long run. Nicolas tries to convince him that this is not the case.
Pierre stakes a sum of money in a casino game. If he wins then he gets back his stake plus the same amount again. If he loses then he loses his stake.
Nicolas says that Pierre can guarantee to win by repeatedly playing the game, even though the probability of winning an individual game is less than 0.5 .
His idea is that Pierre should bet in the first game with a stake of $\pounds 100$. If he wins then he stops, as he will have won $\pounds 100$.
If he loses then he plays again with a stake of $\pounds 200$. If he wins then he has lost $\pounds 100$ and won $\pounds 200$. This gives a total gain of $\pounds 100$, and he stops.
If he loses then he plays again with a stake of $\pounds 400$. If he wins this time he has lost $\pounds 100$ and $\pounds 200$ and won $\pounds 400$. This gives a total gain of $\pounds 100$, and he stops.
Nicolas's advice is that Pierre simply has to continue in this way, doubling his stake every time that he loses, until he eventually wins. Nicolas says that this guarantees that Pierre will win $\pounds 100$.
You are to simulate what might happen if Pierre tries this strategy in a casino game in which the probability of him winning an individual game is 0.4 , and in which he has $\pounds 1000$ available.\\
(i) Give an efficient rule for using 1-digit random numbers to simulate the outcomes of individual games, given that the probability of Pierre winning an individual game is 0.4 .\\
(ii) Explain why at most three random digits are needed for one simulation of Nicolas's strategy, given that Pierre is starting with $\pounds 1000$.\\
(iii) Simulate five applications of Nicolas's strategy, using the five sets of three 1-digit random numbers in your answer book.\\
(iv) Summarise the results of your simulations, giving your mean result.
\hfill \mbox{\textit{OCR MEI D1 2016 Q1 [8]}}