Win or Lose?

A gambler bets half the money in his pocket on the toss of a coin, winning an equal amount for a head and losing his money if the result is a tail. After 2n plays he has won exactly n times. Has he more money than he started with?

Fixing the Odds

You have two bags, four red balls and four white balls. You must put all the balls in the bags although you are allowed to have one bag empty. How should you distribute the balls between the two bags so as to make the probability of choosing a red ball as small as possible and what will the probability be in that case?

Scratch Cards

To win on a scratch card you have to uncover three numbers that add up to more than fifteen. What is the probability of winning a prize?

In a Box

Why do this problem?

This problem offers opportunties to consider different methods of listing systematically. It can be used to introduce or revisit sample space diagrams, and with some students, tree diagrams.

Possible approach

This printable worksheet may be useful: In a Box.

Play the game a few times for real.
"Is this a fair game? How can we be sure?"
Class work in pairs trying to decide and to develop an argument to justify their conjectures.
After about ten minutes, stop to discuss the merits of different arguments and representations. This may be an appropriate point to highlight the benefits of different systematic methods for listing all possibilities, using sample space diagrams and, if pupils have met them before, tree diagrams.

Finding a fair game can become a class activity:
Students help to create a class list of all distinct starting points for the game (for example, four ribbons can be either $1R$ and $3B$ or $2R$ and $2B$). These are written on the board for $3, 4, 5, \ldots$ribbons.
Distribute the task of checking which combinations are fair and record them on the board as pairs of pupils decide.
There are not many solutions that work and if pupils are to notice a pattern amongst the combinations that are fair they may need to consider up to a total of $16$ ribbons.

Spend some time conjecturing about more than $16$ ribbons and test.

Key questions

How can you decide if the game is fair?
How many goes do you think we need to be confident of the likelihood of winning?
Are there efficient systems for recording the different possible combinations?