You are an examiner attempting to devise a scoring system for a multiple choice maths exam. You want to reward correct answers but discourage or penalise guessing.

You will award a fixed number $a$ points for a correct answer and deduct a fixed number $b$ points for an incorrect answer. There are 10 questions on the test, each with 4 possible answers and the students must answer each question. How would you decide on the values of $a$ and $b$? What factors concerning the test might be important to take into account?

Once you have decided upon your scoring system, you give it to the questions setters who invent the questions.

The test will be 30 minutes long and a student's chance of success on each question on the test is shown in the table:

Question | Prob of correct given time on question | ||||

0 minutes (total guess) | 1 minute | 2 minutes | 5 minutes | 10 minutes | |

1 | 25% | 90% | 100% | ||

2 | 25% | 85% | 100% | ||

3 | 25% | 80% | 100% | ||

4 | 25% | 70% | 90% | 100% | |

5 | 25% | 50% | 80% | 100% | |

6 | 25% | 50% | 60% | 80% | 100% |

7 | 25% | 40% | 50% | 80% | 100% |

8 | 25% | 30% | 50% | 80% | 100% |

9 | 25% | 30% | 30% | 50% | 80% |

10 | 25% | 30% | 30% | 40% | 70% |

What strategy would be a good one for the student to adopt given your scoring system?

Suppose now that students were given the option of leaving answers blank, for no score or penalty, how would you choose $a$ and $b$? How would this affect the strategies?

Extension

How would things change if there were more or fewer possibilities for each question?

How would things change if the test had 100 questions instead of 10 questions?