| Exam Board | Edexcel |
|---|---|
| Module | FD2 (Further Decision 2) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Decision tree with EMV |
| Difficulty | Moderate -0.3 This is a straightforward decision tree problem requiring calculation of probabilities for dice outcomes (sum ≥8 is 15/36, doubles is 6/36) and EMV at each decision node. While it involves multiple stages and careful bookkeeping of costs/payoffs, the techniques are standard for D2/FD2 with no novel insight required—slightly easier than average A-level maths overall. |
| Spec | 7.08f Mixed strategies via LP: reformulate as linear programming problem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Tree diagram with at least five end pay-offs, two decision nodes and two chance nodes | M1 | AO 3.3 |
| Correct probabilities for rolling \(\geq 8\) (\(\frac{5}{12}\)) and same number on both dice (\(\frac{7}{12}\), \(\frac{1}{6}\), \(\frac{5}{6}\)) | B1 | AO 1.1b |
| Correct structure for tree diagram with each arc labelled correctly including probabilities; pay-offs: \(25, \frac{-25}{6}, \frac{-25}{6}, \frac{-10}{}, -5, 0\) | A1 | AO 1.1b |
| At least three end pay-offs consistent with stated probabilities; all five attempted | M1 | AO 3.4 |
| Chance nodes attempted with their probabilities | M1 | AO 3.4 |
| cao for chance and decision nodes including double line through inferior option | A1 | AO 1.1b |
| EMV is £0 and Alka should not play the game | B1 | AO 3.2a |
## Question 2:
| Answer/Working | Marks | Guidance |
|---|---|---|
| Tree diagram with at least five end pay-offs, two decision nodes and two chance nodes | M1 | AO 3.3 |
| Correct probabilities for rolling $\geq 8$ ($\frac{5}{12}$) and same number on both dice ($\frac{7}{12}$, $\frac{1}{6}$, $\frac{5}{6}$) | B1 | AO 1.1b |
| Correct structure for tree diagram with each arc labelled correctly including probabilities; pay-offs: $25, \frac{-25}{6}, \frac{-25}{6}, \frac{-10}{}, -5, 0$ | A1 | AO 1.1b |
| At least three end pay-offs consistent with stated probabilities; all five attempted | M1 | AO 3.4 |
| Chance nodes attempted with their probabilities | M1 | AO 3.4 |
| cao for chance and decision nodes including double line through inferior option | A1 | AO 1.1b |
| EMV is £0 and Alka should not play the game | B1 | AO 3.2a |\begin{enumerate}
\item Alka is considering paying $\pounds 5$ to play a game. The game involves rolling two fair six-sided dice. If the sum of the numbers on the two dice is at least 8 , she receives $\pounds 10$, otherwise she loses and receives nothing.
\end{enumerate}
If Alka loses, she can pay a further $\pounds 5$ to roll the dice again. If both dice show the same number then she receives $\pounds 35$, otherwise she loses and receives nothing.\\
(i) Draw a decision tree to model Alka's possible decisions and the possible outcomes.\\
(ii) Determine Alka's optimal EMV and state the optimal strategy indicated by the decision tree.\\
\hfill \mbox{\textit{Edexcel FD2 2021 Q2 [7]}}