| Exam Board | OCR MEI |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modelling and Hypothesis Testing |
| Type | Insurance and risk mitigation decisions |
| Difficulty | Moderate -0.5 This is a straightforward decision tree problem with clear probabilities and outcomes. Parts (i)-(iii) involve standard EMV calculations with given values requiring only arithmetic and tree construction. Part (iv) adds a simple utility weighting but remains mechanical. The multi-part structure and context add length but not conceptual difficulty—this is below average for A-level as it's purely procedural application of decision mathematics algorithms without requiring problem-solving insight or mathematical sophistication. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Decision node leading to two branches: "Vaccinate" and "Don't vaccinate" | B1 | Correct structure with decision node |
| "Vaccinate" branch: chance node with P(disease) = 0.001, loss = £920; P(no disease) = 0.999, loss = £20 | B1 | Correct probabilities and outcomes on vaccinate branch |
| "Don't vaccinate" branch: chance node with P(disease) = 0.02, loss = £900; P(no disease) = 0.98, loss = £0 | B1 | Correct probabilities and outcomes on don't vaccinate branch |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| EMV(Vaccinate) \(= 0.001 \times 920 + 0.999 \times 20 = 0.92 + 19.98 = £20.90\) | M1 A1 | Correct calculation |
| EMV(Don't vaccinate) \(= 0.02 \times 900 + 0.98 \times 0 = £18\) | M1 A1 | Correct calculation |
| Don't vaccinate as £18 < £20.90 | A1 | Correct decision stated |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| If susceptible: EMV(vaccinate) \(= 0.0025 \times 920 + 0.9975 \times 20 = 2.30 + 19.95 = £22.25\) | M1 A1 | |
| If susceptible: EMV(don't vaccinate) \(= 0.05 \times 900 = £45\) | A1 | So vaccinate if susceptible |
| If not susceptible: EMV(vaccinate) \(= 0.0005 \times 920 + 0.9995 \times 20 = 0.46 + 19.99 = £20.45\) | M1 A1 | |
| If not susceptible: EMV(don't vaccinate) \(= 0.01 \times 900 = £9\) | A1 | So don't vaccinate if not susceptible |
| Overall EMV with questionnaire \(= 0.25 \times 22.25 + 0.75 \times 9 = 5.5625 + 6.75 = £12.3125\) | M1 | |
| EMV of questionnaire \(= 18 - 12.3125 = £5.69\) (approx) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Let unpleasantness of disease \(= 1000k\), vaccination \(= k\) | B1 | Setting up utility/unpleasantness scale |
| Vaccinate: expected unpleasantness \(= 0.001 \times 1000k + k = k + k = 2k\) | M1 | |
| Don't vaccinate: expected unpleasantness \(= 0.02 \times 1000k = 20k\) | A1 | |
| Since \(2k < 20k\), Martin should vaccinate | A1 | Correct conclusion |
# Question 1:
## Part (i) [3 marks]
| Answer | Marks | Guidance |
|--------|-------|----------|
| Decision node leading to two branches: "Vaccinate" and "Don't vaccinate" | B1 | Correct structure with decision node |
| "Vaccinate" branch: chance node with P(disease) = 0.001, loss = £920; P(no disease) = 0.999, loss = £20 | B1 | Correct probabilities and outcomes on vaccinate branch |
| "Don't vaccinate" branch: chance node with P(disease) = 0.02, loss = £900; P(no disease) = 0.98, loss = £0 | B1 | Correct probabilities and outcomes on don't vaccinate branch |
## Part (ii) [4 marks]
| Answer | Marks | Guidance |
|--------|-------|----------|
| EMV(Vaccinate) $= 0.001 \times 920 + 0.999 \times 20 = 0.92 + 19.98 = £20.90$ | M1 A1 | Correct calculation |
| EMV(Don't vaccinate) $= 0.02 \times 900 + 0.98 \times 0 = £18$ | M1 A1 | Correct calculation |
| Don't vaccinate as £18 < £20.90 | A1 | Correct decision stated |
## Part (iii) [6 marks]
| Answer | Marks | Guidance |
|--------|-------|----------|
| If susceptible: EMV(vaccinate) $= 0.0025 \times 920 + 0.9975 \times 20 = 2.30 + 19.95 = £22.25$ | M1 A1 | |
| If susceptible: EMV(don't vaccinate) $= 0.05 \times 900 = £45$ | A1 | So vaccinate if susceptible |
| If not susceptible: EMV(vaccinate) $= 0.0005 \times 920 + 0.9995 \times 20 = 0.46 + 19.99 = £20.45$ | M1 A1 | |
| If not susceptible: EMV(don't vaccinate) $= 0.01 \times 900 = £9$ | A1 | So don't vaccinate if not susceptible |
| Overall EMV with questionnaire $= 0.25 \times 22.25 + 0.75 \times 9 = 5.5625 + 6.75 = £12.3125$ | M1 | |
| EMV of questionnaire $= 18 - 12.3125 = £5.69$ (approx) | A1 | |
## Part (iv) [3 marks]
| Answer | Marks | Guidance |
|--------|-------|----------|
| Let unpleasantness of disease $= 1000k$, vaccination $= k$ | B1 | Setting up utility/unpleasantness scale |
| Vaccinate: expected unpleasantness $= 0.001 \times 1000k + k = k + k = 2k$ | M1 | |
| Don't vaccinate: expected unpleasantness $= 0.02 \times 1000k = 20k$ | A1 | |
| Since $2k < 20k$, Martin should vaccinate | A1 | Correct conclusion |
---1 Martin is considering paying for a vaccination against a disease. If he catches the disease he would not be able to work and would lose $\pounds 900$ in income because he would have to stay at home recovering. The vaccination costs $\pounds 20$. The vaccination would reduce his risk of catching the disease during the year from 0.02 to 0.001 .\\
(i) Draw a decision tree for Martin.\\
(ii) Evaluate the EMV of Martin's loss at each node of your tree, and give the action that Martin should take to minimise the EMV of his loss.
Martin can answer a medical questionnaire which will give an estimate of his susceptibility to the disease. If he is found to be susceptible, then his chance of catching the disease is 0.05 . Vaccination will reduce that to 0.0025 . If he is found not to be susceptible, then his chance of catching the disease is 0.01 and vaccination will reduce it to 0.0005 . Historically, $25 \%$ of people are found to be susceptible.\\
(iii) What is the EMV of this questionnaire?
Martin decides not to answer the questionnaire. He also decides that there is more than just his EMV to be considered in deciding whether or not to have the vaccination. The vaccination itself is likely to have side effects, but catching the disease would be very unpleasant. Martin estimates that he would find the effects of the disease 1000 times more unpleasant than the effects of the vaccination.\\
(iv) Analyse which course of action would minimise the unpleasantness for Martin.
\hfill \mbox{\textit{OCR MEI D2 2016 Q1 [16]}}