# Tube Toss

### Task 76 ... Years 4 - 10

#### Summary

Three labelled counters are tossed within a sealed tube. Using a fair ground context and challenging the players to be the stall holder, and therefore needing to determine the criterion for a prize, the task encourages students to explore all the possibilities of this sample space.

#### Materials

• Tube with captured counters numbered 1, 2 and 3 on one side

#### Content

• empirical and theoretical probability
• sample space
• concept of breaking even

#### Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

This task can be approached by experiment (empirically) and through theory. For example, Question 1 explores a win if the 3 lands face up. Students could do 100 trials to guide them in a decision about pay out, or they could consider the possibilities.

In this case the 3 has a one in two chance of occurring because the relevant outcome of the experiment is either a 3 up or not. The other discs don't matter. So, if it costs \$1 per game to play and you paid out \$2 for a win, you would expect no profit or loss. It is important in the development of understanding about chance that the students do carry out the 100 trials to compare the 1 in 2 theory with the short term variation that appears in the experimental approach. For example in one set of 55 trials we carried out, 3 UP scored 25 and 3 DOWN scored 30.

• Would the scores have become more equal after 100 trials?
• Is it more instructive to keep a running record of the changing percentage of each possibility compared to the total number of experiments?
Question 2 investigates a win if either a 1 or a 2 or both show face up. The 3 doesn't matter, so the sample space for the experiment is...

... where Y = 'yes', face up, and N = 'no', face down.

Therefore, in 4 games the player can expect to win 3 times. So, if you charge \$1 to play and expect to make neither profit nor loss, you would pay \$1·331/3 for a win.

 1 2 Y Y Y N N Y N N

Technically this amount isn't possible, because the smallest Australian coin is 5¢, so there is room here to discuss how the game could be promoted so actual currency could be used. Again, it is worth exploring how many trials are enough for the experimental data to fall into line with this theory.

Question 3 involves all three numbers. The player wins with a total of 3 or more. If 0 indicates all discs face down, the sample space for this experiment is:

 0 1 2 3 1+2 1+3 2+3 1+2+3 N N N Y Y Y Y Y

Therefore, in 8 games the player can expect to win 5 times. So, if you charge \$1 to play and expect to make neither profit nor loss, you would pay \$1·60 cents for a win.

An extension to the task is to ask students to design their own rules for a win and explore the related break even payments.

#### Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

It would be useful to have a class set of these tubes if you want to use this task as a whole class lesson. You may be able to purchase empty tubes through a chemist since they are the type of container used to package some pills. However, even if you have only the one copy, you can easily run a class investigation by passing the tube from person to person and making a class record of their experiments. It only takes a moment or two to shake the tube four times, so with 25 students, 100 trials can be completed and recorded quickly.

At this stage, Tube Toss does not have a matching lesson on Maths300.

#### Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

The Tube Toss task is an integral part of:

• MWA Chance & Measurement Years 3 & 4
• MWA Chance & Measurement Years 7 & 8