# Game Show

### Task 162 ... Years 2 - 12

#### Summary

So easy to state and so easy to start, but likely to lead to long debate. The compère of game show has three upside-down opaque cups, one of which has an object underneath. The player points to the one they think has the object and the compère reveals what is underneath another that she knows doesn't have the object. The player has the opportunity to change their guess. The investigative question is Should the player change their mind or stick with the original guess?.

#### Materials

• 3 cups and a small block

#### Content

• concept of proof
• probability calculations
• probability, complementary
• probability, conditional
• probability experiences
• statistics, analysing data
• statistics, collecting & organising data
• statistics, frequency
• statistics, inference

#### Iceberg

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

The task invites students to design an experiment and gather empirical data. However, encourage them to first predict whether it is better to 'stick or swap'. As the cards states, whether to 'stick or swap' has caused significant argument in the past and many mathematicians who were involved in that argument reasoned incorrectly.

The student's experiment will need to try the game with and without change of mind a sufficient number of times and record wins and losses. At least 20 times each way would be necessary before it becomes clear which choice is more likely to give wins. If it isn't clear after 20 experiments of each, then encourage more data to be collected.

Eventually students will find that a win is more likely if the player changes their mind. In fact the statistics move towards a

• 1 in 3 chance of winning if you don't change your mind, and
• 2 in 3 chance of winning if you do change your mind.
For many, even if they believe that changing your mind is better, the expectation is that there is a 50:50 chance of winning if you do change because there are then only two choices, so the actual outcome can be quite counter-intuitive. The challenge now becomes understanding why your chance of winning is doubled if you change your mind and then explaining the result to others.

One way to explain is:

1. If the player decides not to change their mind, their win or loss is decided by their first choice, because they will not change regardless of what the compère reveals. Their first choice was a random one out of three, hence a probability of 1/3 of winning.
2. If the player decides to change their mind their second choice has to be considered. Their first choice could have been correct or not, so let's break this into smaller parts:
• 1st choice correct ... then both remaining cups have no prize and whichever the compère reveals the player will change their mind, be forced to the other one and lose.
• 1st choice not correct ... then one remaining cup is correct and the compère won't reveal that one, hence the player will have be forced to take the winning one because it will be the only one left after the reveal.
So, if the player chooses to not change their mind and has chosen a not correct cup the first time they must win! So, the chances of winning if you change your mind must be the same as the chances of selecting a not correct cup the first time, which is 2 out of 3 or 2/3.

#### 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's easy to get opaque plastic cups or an alternative so that this investigation can be run with the whole class and a small block, or dice, makes a perfect 'prize'. Then an outline of the lesson would be:

1. Set up a demonstration situation with one student as a 'contestant'. All students will have to close their eyes while you hide the prize.
2. Ask the student to choose, then explain that they now have an opportunity to change their mind. Do they want to 'change their mind' or not?
3. Ask the rest of the class to vote on what they would do and record their votes.
4. How could we find out the correct answer? - run an experiment.
5. Each group plays 10 games where they DO NOT change their mind - record number of wins.
6. Then each group plays 10 games where they ALWAYS change mind - record number of wins.
7. Compare the two sets of results on the board - what do they show?
8. Use the Maths300 computer program (if available) or continue gathering further empirical evidence until the chances of a win for each of 'stick or swap' becomes clear.
9. Try to logically 'explain' why the empirical result must be true.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 60, Game Show, which includes software allowing many trials to be run very quickly.

#### 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 Game Show task is an integral part of:

• MWA Chance & Measurement Years 5 & 6
• MWA Chance & Measurement Years 9 & 10

The Game Show lesson is an integral part of:

• MWA Chance & Measurement Years 9 & 10