| Exam Board | Edexcel |
|---|---|
| Module | FD2 (Further Decision 2) |
| Year | 2024 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Decision tree with EMV |
| Difficulty | Standard +0.3 This is a straightforward application of decision tree methodology with EMV calculations. Students must compute expected values by multiplying probabilities with costs (travel + time costs), then compare options. Part (c) adds a simple utility function transformation. The arithmetic is tedious but the conceptual demand is low—it's a standard textbook exercise requiring careful calculation rather than problem-solving insight. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06c Working with constraints: algebra and ad hoc methods |
| Cost of travel option | Journey time (in hours) without delays | Probability of a 1-hour delay | Probability of a 2-hour delay | Probability of a 3-hour delay | Probability of a 24-hour delay | |
| Plane | £200 | 3 | 0.09 | 0.05 | 0 | 0.03 |
| Train | £130 | 5 | 0.07 | 0.03 | 0 | 0 |
| Coach | £70 | 6 | 0.15 | 0.1 | 0.05 | 0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Tree diagram with at least eight end pay-offs, one decision node and three chance nodes | M1 | Condone missing triangles from end pay-offs and incorrect shapes for decision/chance nodes |
| Correct structure with non-zero probability 11 arcs labelled correctly (including probabilities) | A1 | Correct structure of tree diagram |
| At least three end pay-offs consistent with stated probabilities (must include ticket price, cost for travel time and cost for delay); at least eight attempted | M1 | May be implied by correct values |
| All eleven end pay-offs correct including triangles, no incorrect extras | A1 | Condone if not fully simplified; values may be negative as costs |
| All three chance nodes attempted with their probabilities | M1 | — |
| CAO for chance and decision nodes including double line through inferior options | A1 | Must have correct shapes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Travel option is Train | dB1 | Deduction of correct travel option (dependent on all method marks in (a)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Utility values are \(7.2427\ldots,\ 7.2820\ldots,\ 7.3282\ldots\) | M1 A1 | M1: at least one correct; A1: at least two correct |
| Therefore the travel option with best expected utility is Plane | A1 | Correct travel option (Plane) together with all three correct values (to at least 3 sf – rounded or truncated) |
# Question 5:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Tree diagram with at least eight end pay-offs, one decision node and three chance nodes | M1 | Condone missing triangles from end pay-offs and incorrect shapes for decision/chance nodes |
| Correct structure with non-zero probability 11 arcs labelled correctly (including probabilities) | A1 | Correct structure of tree diagram |
| At least three end pay-offs consistent with stated probabilities (must include ticket price, cost for travel time and cost for delay); at least eight attempted | M1 | May be implied by correct values |
| All eleven end pay-offs correct including triangles, no incorrect extras | A1 | Condone if not fully simplified; values may be negative as costs |
| All three chance nodes attempted with their probabilities | M1 | — |
| CAO for chance and decision nodes including double line through inferior options | A1 | Must have correct shapes |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Travel option is Train | dB1 | Deduction of correct travel option (dependent on all method marks in (a)) |
## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Utility values are $7.2427\ldots,\ 7.2820\ldots,\ 7.3282\ldots$ | M1 A1 | M1: at least one correct; A1: at least two correct |
| Therefore the travel option with best expected utility is Plane | A1 | Correct travel option (Plane) together with all three correct values (to at least 3 sf – rounded or truncated) |
---5. Sebastien needs to make a journey. He can choose between travelling by plane, by train or by coach.
Sebastien knows the exact costs of all three travel options, but he also wants to account for his travel time, including any possible delays.
The cost of Sebastien's time is $\pounds 50$ per hour.\\
The table below shows the costs, the journey times (without delays), and the corresponding probabilities of delays, for each travel option.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
& Cost of travel option & Journey time (in hours) without delays & Probability of a 1-hour delay & Probability of a 2-hour delay & Probability of a 3-hour delay & Probability of a 24-hour delay \\
\hline
Plane & £200 & 3 & 0.09 & 0.05 & 0 & 0.03 \\
\hline
Train & £130 & 5 & 0.07 & 0.03 & 0 & 0 \\
\hline
Coach & £70 & 6 & 0.15 & 0.1 & 0.05 & 0 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item By drawing a decision tree, evaluate the EMV of the total cost of Sebastien's journey for each node of your tree.
\item Hence state the travel option that minimises the EMV of the total cost of Sebastien's journey.
\item A cube root utility function is applied to the total costs of each option. Determine the travel option with the best expected utility and state the corresponding value.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FD2 2024 Q5 [10]}}