| Exam Board | Edexcel |
|---|---|
| Module | FD2 (Further Decision 2) |
| Year | 2020 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Decision tree with utility |
| Difficulty | Standard +0.3 This is a standard decision tree problem with EMV calculation followed by expected utility comparison. While it requires careful arithmetic with probabilities and exponential utility functions, it follows a routine algorithmic approach taught directly in FD2 with no novel problem-solving or insight required. The two-part structure and utility function application make it slightly above average difficulty for A-level, but it remains a textbook exercise. |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space2.04a Discrete probability distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| EMV for A is \(0.6(350) + 0.4(-140) = 154\) | M1 | Correct method for calculating EMV for either A, B or C |
| EMV for B is \(0.75(260) + 0.25(-190) = 147.5\) | A1 | Correct values of EMV for A, B and C |
| EMV for C is \(0.8(220) + 0.2(-230) = 130\) | ||
| The optimal EMV is £154, which makes option A the best choice using the EMV criterion | A1 | Correct deduction of optimal EMV (dependent on all three correct EMVs) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(u(350) = 0.5831...\), \(u(-140) = -0.4191...\) | M1 | Uses correct utility function to replace each pay-off with corresponding utility |
| \(u(260) = 0.4780...\), \(u(-190) = -0.6080...\) | ||
| \(u(220) = 0.4231...\), \(u(-230) = -0.7771...\) | ||
| A is \(0.6(0.583...) + 0.4(-0.419...) = 0.1823...\) | DM1 | Calculate all three expected utilities using correct probability values from (a) |
| B is \(0.75(0.477...) + 0.25(-0.608...) = 0.2065...\) | A1 | At least 2 expected utilities correct (to at least 2 d.p.) |
| C is \(0.8(0.423...) + 0.2(-0.777...) = 0.1830...\) | ||
| The optimal expected utility is 0.206 utils, which makes option B the best choice using expected utility as the criterion | A1 | Correct deduction of optimal expected utility |
# Question 2:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| EMV for A is $0.6(350) + 0.4(-140) = 154$ | M1 | Correct method for calculating EMV for either A, B or C |
| EMV for B is $0.75(260) + 0.25(-190) = 147.5$ | A1 | Correct values of EMV for A, B and C |
| EMV for C is $0.8(220) + 0.2(-230) = 130$ | | |
| The optimal EMV is £154, which makes option A the best choice using the EMV criterion | A1 | Correct deduction of optimal EMV (dependent on all three correct EMVs) |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $u(350) = 0.5831...$, $u(-140) = -0.4191...$ | M1 | Uses correct utility function to replace each pay-off with corresponding utility |
| $u(260) = 0.4780...$, $u(-190) = -0.6080...$ | | |
| $u(220) = 0.4231...$, $u(-230) = -0.7771...$ | | |
| A is $0.6(0.583...) + 0.4(-0.419...) = 0.1823...$ | DM1 | Calculate all three expected utilities using correct probability values from (a) |
| B is $0.75(0.477...) + 0.25(-0.608...) = 0.2065...$ | A1 | At least 2 expected utilities correct (to at least 2 d.p.) |
| C is $0.8(0.423...) + 0.2(-0.777...) = 0.1830...$ | | |
| The optimal expected utility is 0.206 utils, which makes option B the best choice using expected utility as the criterion | A1 | Correct deduction of optimal expected utility |
---2. Jenny can choose one of three options, A, B or C, when playing a game. The profit, in pounds, associated with each outcome and their corresponding probabilities are shown on the decision tree in Figure 1.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{e0fc09f9-06ea-4528-a2de-f409112d85cc-03_947_1319_349_374}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
\begin{enumerate}[label=(\alph*)]
\item Calculate the optimal EMV to determine Jenny's best course of action. You must make your working clear.
For a profit of $\pounds x$, Jenny's utility is given by $1 - \mathrm { e } ^ { - \frac { x } { 400 } }$
\item Using expected utility as the criterion for the best course of action, determine what Jenny should do now to maximise her profit. You must make your working clear.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FD2 2020 Q2 [7]}}