### Why do this problem?

This
problem offers a great opportunity for talking about chance.
Students can use their intuition to rank the options in order, and
then model the situation and manipulate the resulting fractions.
Finally, there is a chance to discuss whether the models used are
appropriate and how this might affect their answer.

### Possible approach

Present the problem, and explain that students are being asked
to try their luck without having access to anything more
sophisticated than paper and pencil. Allow some time for pairs or
small groups to discuss the different options, making sure they
understand what they mean. Each group should come up with which
option they think gives them the best chance or the worst chance of
winning, with some justification for why they believe it.

Now give the groups time to write down the calculation they
would use to represent each situation, encouraging use of fractions
rather than decimals for easier working without calculators.

Check that groups have modelled each situation appropriately.
This is a good opportunity for some discussion about the more
subjective options - those which involve a ranking based on
preference - and how the probability would change given more
information about the people who put them in order.

Once everyone has a model for each option, groups can start to
compare different options. Some comparisons are easier than others,
so encourage use of prime factorisation to compare the size of the
denominators. Laws of indices can be useful to compare for example
$(\frac{1}{2})^{12}$ with $(\frac{1}{10})^4$.

Once comparisons have been made, students should compare their
answers with their intuition about which was best, and consider
whether they would try to win the holiday or keep their money!
Another discussion point could be how much the prize should be
worth in order for the organisers to make money from the
enterprise.

### Key questions

Which game do you think is easiest to win?

How can we compare two fractions with different denominators?

### Possible extension

Come up with other scenarios with similar probabilities.

The
Better Bet is another problem about choosing which game to play
in order to maximise winnings.

### Possible support

Allow some calculator use so the focus is on the probability
calculations without the comparisons of fractions getting in the
way.