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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?

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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.

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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?

Do You Feel Lucky?

Stage: 3 Challenge Level: Challenge Level:1

Why do this problem?

This problem is one of a set of problems about probability and uncertainty. Intuition can often let us down when we meet probability in real life contexts; this problem has been designed to provoke discussions that challenge commonly held misconceptions such as the Gambler's Fallacy.

Possible approach

This printable worksheet may be useful: Do You Feel Lucky.

Hand out this worksheet (Word, PDF) with the statements from the problem, and give everyone time to read them through and decide for themselves whether they think the advice is good or not.
Then arrange the class into pairs, and ask half the pairs to come up with arguments in favour and half to come up with arguments against following the advice. Once each pair has had time to rehearse their arguments, arrange them in groups of four with one pair arguing in favour and one pair arguing against, until they reach a consensus.
Next, bring the class together for a discussion about how they would respond to each piece of advice and the mathematics they used to justify their decisions.
Finally, ask everyone to read through the statements on their own again, to see if they have changed their original views about any of the advice. Those who have changed their minds could explain to the rest of the class which arguments they found particularly persuasive.
In the statements, it has been deliberately left ambiguous whether coins are fair or not and so on, so it may be that there is no "right" answer that can be agreed on. The important point is for learners to discuss intelligently the probability in the situations and challenge some of the popular misconceptions that arise.  

Key questions 

Are future results affected by previous results?
Lots of advice is based on accurate statistical data - does that necessarily mean it is useful advice?

Possible extension

Ask learners to collect over several weeks some examples of probability misconceptions in the media, in school or at home, which could be used to create a classroom display.
The stage 5 problem Discussing Risk and Reward provides more prompts for discussion about probability in the real world.

Possible support

The problem has been structured as a discussion task so that learners can support each other in coming to a better understanding. By allocating a view for each pair to argue, it allows those who hold these misconceptions the chance to freely explore them without fear of ridicule.