6 Sam is playing a computer game in which he is trying to drive a car in different road conditions. He chooses a car and the computer decides the road conditions. The points scored by Sam are shown in the table.
| Road Conditions |
| \cline { 2 - 5 } | | \(\boldsymbol { C } _ { \mathbf { 1 } }\) | \(\boldsymbol { C } _ { \mathbf { 2 } }\) | \(\boldsymbol { C } _ { \mathbf { 3 } }\) |
| \cline { 2 - 5 } | \(\boldsymbol { S } _ { \mathbf { 1 } }\) | - 2 | 2 | 4 |
| \cline { 2 - 5 }
Sam's Car | \(\boldsymbol { S } _ { \mathbf { 2 } }\) | 2 | 4 | 5 |
| \cline { 2 - 5 } | \(\boldsymbol { S } _ { \mathbf { 3 } }\) | 5 | 1 | 2 |
| \cline { 2 - 5 } | | | | |
| \cline { 2 - 5 } |
Sam is trying to maximise his total points and the computer is trying to stop him.
- Explain why Sam should never choose \(S _ { 1 }\) and why the computer should not choose \(C _ { 3 }\).
- Find the play-safe strategies for the reduced 2 by 2 game for Sam and the computer, and hence show that this game does not have a stable solution.
- Sam uses random numbers to choose \(S _ { 2 }\) with probability \(p\) and \(S _ { 3 }\) with probability \(1 - p\).
- Find expressions for the expected gain for Sam when the computer chooses each of its two remaining strategies.
- Calculate the value of \(p\) for Sam to maximise his total points.
- Hence find the expected points gain for Sam.
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Advanced Level Examination}
\section*{MATHEMATICS
Unit Decision 2}
MD02
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Wednesday 18 January 20061.30 pm to 3.00 pm
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