| Exam Board | OCR MEI |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Decision tree with EMV |
| Difficulty | Moderate -0.8 This is a standard D2 decision tree question with straightforward probability calculations and EMV computations. Parts (i)-(iii) involve routine tree construction and expected value calculations, while part (iv) requires solving a simple equation. The utility function application in part (iii) is mechanical. No novel problem-solving insight is required—just systematic application of taught techniques. |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | 3 | 6 |
| 2 | 1 | 1 |
| 2 | 14 | 28 |
| 2 | 1 | 1 |
| 2 | 22 | 44 |
| 2 | 1 | 1 |
| 2 | 15 | 30 |
| 2 | 1 | 1 |
| 2 | 13 | 4 |
| 2 | 5 | 4 |
Question 2:
2
1 (a) Ajoke has it that army recruits used to be instructed: “If it moves, salute it. If it doesn’t move,
paint it.”
Assume that this instruction has been carried out completely in the local universe, so that
everything that doesn't move has been painted.
(i) Arecruit encounters something which is not painted. What should he do, and why? [3]
(ii) Arecruit encounters something which is painted. Do we know what he or she should do?
Justify your answer. [3]
(b) Use a truth table to prove (((m fi s) (cid:1) (~ m fi p)) (cid:1) ~ p) fi s. [6]
(c) You are given the following two rules.
1 (a fi b) ¤ (~ b fi ~ a)
2 (x (cid:1)(x fi y)) fi y
Use Boolean algebra to prove that (((m fi s) (cid:1) (~ m fi p)) (cid:1) ~ p) fi s. [4]
2 Bill is at a horse race meeting. He has £2 left with two races to go. He only ever bets £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 £1 and the horse wins, then Bill will
receive back his £1 plus £1 winnings. If Bill’s horse does not win then Bill will lose his £1.
(i) 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. [8]
Assume that in each race the probability of Bill’s horse winning is the same, and that it has value p.
(ii) Find Bill’s EMV when
(A) p (cid:1) 0.6,
(B) p (cid:1) 0.4.
Give his best course of action in each case. [5]
(iii) Suppose that Bill uses the utility function utility (cid:1) (money)x, to decide whether or not to bet
£1 on one race. Show that, with p (cid:1) 0.4, Bill will not bet if x (cid:1) 0.5, but will bet if x (cid:1) 1.5.
[3]
2 | 3 | 6 | 7 | 2
2 | 1 | 1 | 3 | 4
2 | 14 | 28 | 15 | 27
2 | 1 | 1 | 3 | 3
2 | 22 | 44 | 20 | 5 | 23
2 | 1 | 1 | 3 | 4 | 5
2 | 15 | 30 | 6 | 2 | 6 | 23
2 | 1 | 1 | 3 | 4 | 5 | 1
2 | 13 | 4 | 5 | 2 | 6 | 14
2 | 5 | 4 | 4 | 4 | 5 | 5Karl is considering investing in a villa in Greece. It will cost him 56000 euros (€ 56000). His alternative is to invest his money, £35000, in the United Kingdom.
He is concerned with what will happen over the next 5 years. He estimates that there is a 60% chance that a house currently worth € 56000 will appreciate to be worth € 75000 in that time, but that there is a 40% chance that it will be worth only € 55000.
If he invests in the United Kingdom then there is a 50% chance that there will be 20% growth over the 5 years, and a 50% chance that there will be 10% growth.
\begin{enumerate}[label=(\roman*)]
\item Given that £1 is worth € 1.60, draw a decision tree for Karl, and advise him what to do, using the EMV of his investment (in thousands of euros) as his criterion. [4]
\end{enumerate}
In fact the £/€ exchange rate is not fixed. It is estimated that at the end of 5 years, if there has been 20% growth in the UK then there is a 70% chance that the exchange rate will stand at 1.70 euros per pound, and a 30% chance that it will be 1.50. If growth has been 10% then there is a 40% chance that the exchange rate will stand at 1.70 and a 60% chance that it will be 1.50.
\begin{enumerate}[label=(\roman*), start=2]
\item Produce a revised decision tree incorporating this information, and give appropriate advice. [3]
\end{enumerate}
A financial analyst asks Karl a number of questions to determine his utility function. He estimates that for x in cash (in thousands of euros) Karl's utility is $x^{0.5}$, and that for y in property (in thousands of euros), Karl's utility is $y^{0.75}$.
\begin{enumerate}[label=(\roman*), start=3]
\item Repeat your computations from part (ii) using utility instead of the EMV of his investment. Does this change your advice? [3]
\item Using EMVs, find the exchange rate (number of euros per pound) which will make Karl indifferent between investing in the UK and investing in a villa in Greece. [2]
\item Show that, using Karl's utility function, the exchange rate would have to drop to 1.277 euros per pound to make Karl indifferent between investing in the UK and investing in a villa in Greece. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI D2 Q2 [16]}}