OCR MEI D1 2012 June — Question 5 16 marks

Exam BoardOCR MEI
ModuleD1 (Decision Mathematics 1)
Year2012
SessionJune
Marks16
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicModelling and Hypothesis Testing
TypeMulti-stage probability simulation
DifficultyModerate -0.3 This is a straightforward simulation exercise requiring basic probability concepts and following mechanical rules. Students must assign random digits to outcomes proportional to probabilities (e.g., 1 out of 9 bags → digit 0 represents single bag, 1-8 represent double bags), then trace through given random numbers. While multi-stage, it requires no sophisticated mathematical insight—just careful bookkeeping and understanding that simulation estimates probabilities through repeated trials. Easier than average A-level as it's procedural rather than problem-solving.
Spec2.01c Sampling techniques: simple random, opportunity, etc2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space
5 Each morning I reach into my box of tea bags and, without looking, randomly choose a bag. The bags are manufactured in pairs, which can be separated along a perforated line. So when I choose a bag it might be attached to another, in which case I have to separate them and return the other bag to the box. Alternatively, it might be a single bag, having been separated on an earlier day. I only use one tea bag per day, and the box always gets thoroughly shaken during the day as things are moved around in the kitchen. You are to simulate this process, starting with 5 double bags and 0 single bags in the box. You are to use single-digit random numbers in your simulation.
  1. On day 2 there will be 4 double bags and 1 single bag in the box, 9 bags in total. Give a rule for simulating whether I choose a single bag or a double bag, assuming that I am equally likely to choose any of the 9 bags. Use single-digit random numbers in your simulation rule.
  2. On day 3 there will either be 4 double bags or 3 double bags and 2 single bags in the box. Give a rule for simulating what sort of bag I choose in the second of these cases. Use single-digit random numbers in your simulation rule.
  3. Using the random digits in your answer book, simulate what happens on days 2,3 and 4 , briefly explaining your simulations. Give an estimate of the probability that I choose a single bag on day 5 .
  4. Using the random digits in your answer book, carry out 4 more simulations and record the results.
  5. Using your 5 simulations, estimate the probability that I choose a single bag on day 5 .
    [0pt] [Question 6 is printed overleaf.]
Question 5:
Part (i)
AnswerMarks Guidance
AnswerMarks Guidance
e.g. \(0\text{–}7 \rightarrow\) double; \(8 \rightarrow\) single; \(9\) reject and re-drawM1, A1 Rejection can be implied. Correct proportions.
[2]
Part (ii)
AnswerMarks Guidance
AnswerMarks Guidance
e.g. \(0\text{–}5 \rightarrow\) double; \(6,7 \rightarrow\) single; \(8,9\) reject and re-drawM1, A1 Rejection can be implied. Ignore rule for \((4,0)\). Correct proportions.
[2]
Part (iii)
AnswerMarks Guidance
AnswerMarks Guidance
Day 1: doubles=5, singles=0; Day 2: doubles=4, singles=1, RN=5 → doubleM1, A1 Allow 5 shown as used on RN list. Selection.
Day 3: doubles=3, singles=2, RN=9,4 → doubleM1, A1 Must show RN(s) explicitly; new scenario seen explicitly, not implied by day 4 rule. Follow candidate who manages correctly from \((4,1)\) to \((4,0)\); gains M1 if correctly goes to \((3,1)\) on day 4, with A1 if shows no simulation needed.
Day 4: doubles=2, singles=3, RN=0 → doubleM1, A1 A correct day 4 rule; selection and new scenario. Rule must be seen; needs RN explicit. Allow new scenario if seen in subsequent probability calculation.
Day 5: doubles=1, singles=4
Probability of drawing a single bag on day 5 is now \(\frac{4}{6}\)M1, A1 Denominator \(= 6\); numerator. Can be implied by \(\frac{2}{3}\) or \(\frac{1}{3}\) if correct for their simulation.
[8]
Question 5:
Part (iv)
AnswerMarks Guidance
AnswerMarks Guidance
4 simulations, each ending with 6 bagsM1 Condone one slip. Condone simulating at \((4,0)\) if correctly done.
all scenarios correctA1 6 bags can be implied by probs of thirds or sixths.
[2]
Part (v)
AnswerMarks Guidance
AnswerMarks Guidance
Either averaging correct probabilities or sum of singles\(/30\)M1 Correct computation, but allow 1 slip or omission.
A1Correct answer for their simulations.
[2] 
# Question 5:

## Part (i)

| Answer | Marks | Guidance |
|--------|-------|----------|
| e.g. $0\text{–}7 \rightarrow$ double; $8 \rightarrow$ single; $9$ reject and re-draw | M1, A1 | Rejection can be implied. Correct proportions. |
| **[2]** | | |

## Part (ii)

| Answer | Marks | Guidance |
|--------|-------|----------|
| e.g. $0\text{–}5 \rightarrow$ double; $6,7 \rightarrow$ single; $8,9$ reject and re-draw | M1, A1 | Rejection can be implied. Ignore rule for $(4,0)$. Correct proportions. |
| **[2]** | | |

## Part (iii)

| Answer | Marks | Guidance |
|--------|-------|----------|
| Day 1: doubles=5, singles=0; Day 2: doubles=4, singles=1, RN=5 → double | M1, A1 | Allow 5 shown as used on RN list. Selection. |
| Day 3: doubles=3, singles=2, RN=9,4 → double | M1, A1 | Must show RN(s) explicitly; new scenario seen explicitly, not implied by day 4 rule. Follow candidate who manages correctly from $(4,1)$ to $(4,0)$; gains M1 if correctly goes to $(3,1)$ on day 4, with A1 if shows no simulation needed. |
| Day 4: doubles=2, singles=3, RN=0 → double | M1, A1 | A correct day 4 rule; selection and new scenario. Rule must be seen; needs RN explicit. Allow new scenario if seen in subsequent probability calculation. |
| Day 5: doubles=1, singles=4 | | |
| Probability of drawing a single bag on day 5 is now $\frac{4}{6}$ | M1, A1 | Denominator $= 6$; numerator. Can be implied by $\frac{2}{3}$ or $\frac{1}{3}$ if correct for their simulation. |
| **[8]** | | |

# Question 5:

## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| 4 simulations, each ending with 6 bags | M1 | Condone one slip. Condone simulating at $(4,0)$ if correctly done. |
| all scenarios correct | A1 | 6 bags can be implied by probs of thirds or sixths. |
| | [2] | |

## Part (v)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Either averaging correct probabilities or sum of singles$/30$ | M1 | Correct computation, but allow 1 slip or omission. |
| | A1 | Correct answer for their simulations. |
| | [2] | |

---
5 Each morning I reach into my box of tea bags and, without looking, randomly choose a bag. The bags are manufactured in pairs, which can be separated along a perforated line. So when I choose a bag it might be attached to another, in which case I have to separate them and return the other bag to the box. Alternatively, it might be a single bag, having been separated on an earlier day.

I only use one tea bag per day, and the box always gets thoroughly shaken during the day as things are moved around in the kitchen.

You are to simulate this process, starting with 5 double bags and 0 single bags in the box. You are to use single-digit random numbers in your simulation.\\
(i) On day 2 there will be 4 double bags and 1 single bag in the box, 9 bags in total. Give a rule for simulating whether I choose a single bag or a double bag, assuming that I am equally likely to choose any of the 9 bags. Use single-digit random numbers in your simulation rule.\\
(ii) On day 3 there will either be 4 double bags or 3 double bags and 2 single bags in the box. Give a rule for simulating what sort of bag I choose in the second of these cases. Use single-digit random numbers in your simulation rule.\\
(iii) Using the random digits in your answer book, simulate what happens on days 2,3 and 4 , briefly explaining your simulations. Give an estimate of the probability that I choose a single bag on day 5 .\\
(iv) Using the random digits in your answer book, carry out 4 more simulations and record the results.\\
(v) Using your 5 simulations, estimate the probability that I choose a single bag on day 5 .\\[0pt]
[Question 6 is printed overleaf.]

\hfill \mbox{\textit{OCR MEI D1 2012 Q5 [16]}}
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