### The Lady or the Lions

The King showed the Princess a map of the maze and the Princess was allowed to decide which room she would wait in. She was not allowed to send a copy to her lover who would have to guess which path to follow. Which room should she wait in to give her lover the greatest chance of finding her?

### A Dicey Paradox

Four fair dice are marked differently on their six faces. Choose first ANY one of them. I can always choose another that will give me a better chance of winning. Investigate.

### Nines and Tens

Explain why it is that when you throw two dice you are more likely to get a score of 9 than of 10. What about the case of 3 dice? Is a score of 9 more likely then a score of 10 with 3 dice?

# Misunderstanding Randomness

#### Why do this problem?

All work on probability is based on ideas of randomness, an idea which has precise mathematical meaning, while being informally used in everyday life. A discussion of tricky ideas should challenge students' understanding.

#### Possible approach

Stand rolling a die, or shuffling some cards as the students enter, or while the discussion starts. If you have an interactive whiteboard, you could leave a slideshow running the numbers1-10 set to shuffle and loop. Ask students for initial ideas of what randomness means, and a show of hands for how many already understand it.

Put students into small groups (randomly?) and ask them to compose one sentence that explains randomness. Organise a 'random relay' - give each group a slip of paper with one of the statements from the problem on it. They must decide whether they agree with it or not, and settle on the main arguments in case they are called on to argue with a group with the opposite opinion. They then get a new statement to work on. Make it very clear that the whole group is responsible for the answers, and that any of them might get called to explain. Tell them that the points available are 10 for every right answer, -20 for each wrong one, (to promote certainty above speed).

Select a few items for debate where groups have reached different final answers. Either arrange for representatives of the groups to meet and convince each other, or arrange a public debate with one champion from either view, and then questions from the floor.

The final scores for each group might reflect the general misunderstanding of probability in the general population, the ideas are difficult, but we can make sure that we understand the basics - reiterate them from the board.

#### Key questions

How does your justification relate to the original statements made about randomness? (perhaps worth displaying these on the board throughout the lesson)

#### Possible extension

Focus on clarity of explanation and attempt to get good answers on all statements.

#### Possible support

Students could start by working with interactivities like Interactive Spinners and Flippin' Discs in order to establish the following key ideas:

- If something is random, you can't ever work out what the next one will be
- Even if it has been done lots of times, the next one could still be anything
- When it's been done lots of times the overall results are very predictable