| Exam Board | OCR MEI |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modelling and Hypothesis Testing |
| Type | Sequential betting and gambling decisions |
| Difficulty | Standard +0.3 This is a straightforward decision tree problem with simple probability calculations and EMV comparisons. Part (i) requires drawing a standard tree, part (ii) involves basic expected value calculations with given probabilities, and part (iii) applies a utility function with direct substitution. All techniques are routine for D2 students with no novel problem-solving required. |
| Spec | 7.05a Critical path analysis: activity on arc networks7.05b Forward and backward pass: earliest/latest times, critical activities7.05d Latest start and earliest finish: independent and interfering float7.05e Cascade charts: scheduling and effect of delays |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Correct first decision node (bet/don't bet, race 1) | B1 | |
| Correct chance nodes after betting with win/lose branches | B1 | |
| Correct second decision nodes (bet/don't bet, race 2) for each outcome | B2 | |
| Correct payoffs: £4 (W,W), £2 (W,L or W,nb), £2 (L,W or nb,W), £0 (L,L), etc. | B3 | All 8+ terminal values correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| A) \(p=0.6\): EMV calculated correctly, best strategy identified | B2 | e.g. bet both races, EMV = £2.48 or equivalent |
| B) \(p=0.4\): EMV calculated correctly, best strategy identified | B2 | e.g. don't bet, EMV = £2 or equivalent |
| Correct best course of action stated for each | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Utility of betting: \(p \cdot (money+1)^x + (1-p)(money-1)^x\) compared to \((money)^x\) | M1 | |
| With \(p=0.4\), \(x=0.5\): \(0.4\sqrt{3} + 0.6\sqrt{1} = 0.693+0.6 = 1.293 < \sqrt{2} \approx 1.414\), don't bet | A1 | |
| With \(p=0.4\), \(x=1.5\): \(0.4(3)^{1.5}+0.6(1)^{1.5} = 2.079+0.6=2.679 > (2)^{1.5}\approx 2.828\)... correct evaluation showing bet | A1 | Accept correct numerical comparisons |
# Question 2:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct first decision node (bet/don't bet, race 1) | B1 | |
| Correct chance nodes after betting with win/lose branches | B1 | |
| Correct second decision nodes (bet/don't bet, race 2) for each outcome | B2 | |
| Correct payoffs: £4 (W,W), £2 (W,L or W,nb), £2 (L,W or nb,W), £0 (L,L), etc. | B3 | All 8+ terminal values correct |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| **A)** $p=0.6$: EMV calculated correctly, best strategy identified | B2 | e.g. bet both races, EMV = £2.48 or equivalent |
| **B)** $p=0.4$: EMV calculated correctly, best strategy identified | B2 | e.g. don't bet, EMV = £2 or equivalent |
| Correct best course of action stated for each | B1 | |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Utility of betting: $p \cdot (money+1)^x + (1-p)(money-1)^x$ compared to $(money)^x$ | M1 | |
| With $p=0.4$, $x=0.5$: $0.4\sqrt{3} + 0.6\sqrt{1} = 0.693+0.6 = 1.293 < \sqrt{2} \approx 1.414$, don't bet | A1 | |
| With $p=0.4$, $x=1.5$: $0.4(3)^{1.5}+0.6(1)^{1.5} = 2.079+0.6=2.679 > (2)^{1.5}\approx 2.828$... correct evaluation showing bet | A1 | Accept correct numerical comparisons |
---2 Bill is at a horse race meeting. He has $\pounds 2$ left with two races to go. He only ever bets $\pounds 1$ at a time. For each race he chooses a horse and then decides whether or not to bet on it. In both races Bill's horse is offered at "evens". This means that, if Bill bets $\pounds 1$ and the horse wins, then Bill will receive back his $\pounds 1$ plus $\pounds 1$ winnings. If Bill's horse does not win then Bill will lose his $\pounds 1$.
\begin{enumerate}[label=(\roman*)]
\item Draw a decision tree to model this situation. Show Bill's payoffs on your tree, i.e. how much money Bill finishes with under each possible outcome.
Assume that in each race the probability of Bill's horse winning is the same, and that it has value $p$.
\item Find Bill's EMV when\\
(A) $p = 0.6$,\\
(B) $p = 0.4$.
Give his best course of action in each case.
\item Suppose that Bill uses the utility function utility $= ( \text { money } ) ^ { x }$, to decide whether or not to bet $\pounds 1$ on one race. Show that, with $p = 0.4$, Bill will not bet if $x = 0.5$, but will bet if $x = 1.5$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI D2 2007 Q2 [16]}}