Fair Games

Getting Started

Introduce the game, “Marble snap” to the class. Place three red and one blue marble in one bag and two red and two blue in the other bag. Invite a student to pull a marble from each bag and predict whether they will be the same (snap) or different. If the marbles are the same they win. Replace the marbles after each trial. Repeat this with a few more students.
Do you think the game is fair? Yes it is, surprisingly.
What do we mean by a fair game?
A fair game is one in which there is an equal chance of winning or losing.
How could we tell if the game is fair?
There are two essential approaches – experimental and theoretical. Accept and list suggestions from the class. These may include:
Do an experiment.
Do a long run experiment.
List the possible events.
List the possible outcomes.
Use informal diagrams to list the possible outcomes.
Use a grid to list the possible outcomes.
Use a tree diagram to list the possible outcomes.

Suggest that carrying out an experiment is the best place to start. Tell the class that you will ask a number of students to pick marbles from the bags to try to get snap.
How will we record the results? A tally chart with win and lose is the simplest approach.
Carry out 10 trials and record on the board.
What do you think of these results? Do they mean that the game is fair? What would happen if we did the experiment again? How could we improve the experiment?
Hopefully the suggestion of a larger experiment to obtain the long-run frequency will be made.

Organise the students into pairs or small groups so that each student can conduct 10 trials. While the students are working ensure that they are conducting the trials correctly, using replacement of the marbles in the correct bags. Collect in the results from the class and summarise on the board.

What do you think of these results? Do they mean that the game is fair? What would happen if we did the experiment again? The long-run frequency obtained by this experiment should not vary significantly in other similar experiments.

Return to the students’ other suggestions of determining whether the game is fair. The important ones to work through are use of grids and use of tree diagrams to list all possible outcomes. However which order you work through them should be determined by which ideas the students hold.

1. Grid method
   What colour might the marble in the first bag be?
   Does it have the same chance of being red or blue?
   What are the possibilities of what could happen when we pull a marble from the first bag? It is important that they list red, red, red and blue as the possible outcomes.
   If I pulled a red marble from the first bag what might happen when I pull a marble from the second bag? It is important that they list red, red, blue and blue as the possible outcomes.
   If I pulled a blue marble from the first bag what might happen when I pull a marble from the second bag? It is important that they list red, red, blue and blue as the possible outcomes.
   Let us list these outcomes on a grid.

How many of these possible outcomes are ‘snap’?
How many of these possible outcomes are not ‘snap’?
What does this tell us about the fairness of the game?

2. Tree diagram method
   What colour might the marble in the first bag be?
   Does it have the same chance of being red or blue?
   What are the possibilities of what could happen when we pull a marble from the first bag? It is important that they list red, red, red and blue as the possible outcomes
   Draw these outcomes as the four branches of the first level of the tree diagram.

Take each branch in turn. If I pulled a red marble from the first bag what might happen when I pull a marble from the second bag? It is important that they list red, red, blue and blue as the possible outcomes.Draw these four branches as the second level of the tree diagram. Repeat this for the other two red branches.
If I pulled a blue marble from the first bag what might happen when I pull a marble from the second bag?It is important that they list red, red, blue and blue as the possible outcomes. Draw these four branches as the second level of the tree diagram.

Now let us list the outcomes.

How many of these possible outcomes are ‘snap’?
How many of these possible outcomes are not ‘snap’?
What does this tell us about the fairness of the game?

3. Sophisticated tree diagram approach
   It is not suggested that this approach is used unless students have proposed it. Even if some students have suggested it, it might be best to treat it as extension material either later or with just some students.
   What colour might the marble in the first bag be?
   Does it have the same chance of being red or blue?
   What is the probability of it being red?
   What is the probability of it being blue?
   Draw these outcomes as the two branches of the first level of the tree diagram, with the probability written above each branch.

Take each branch in turn. If I pulled a red marble from the first bag what might happen when I pull a marble from the second bag? Draw these two branches as the second level of the tree diagram, with the probability written above each branch.
If I pulled a blue marble from the first bag what might happen when I pull a marble from the second bag? Draw these two branches as the second level of the tree diagram, with the probability written above each branch.

Now let us list the outcomes.

How many of these possible outcomes are ‘snap’?
What is the probability of RR, RB, BR and BB?Students need to see the equivalence of this sophisticated tree diagram with the simple one.Trying to teach them to multiply along branches without understanding is counter-productive.
What does this tell us about the fairness of the game?

4. List the possible events
   Students may suggest that the things that could happen are:
   RR
   RB
   BB
   BR
   If they suggest that this means the game is fair, question But what is the probability of RR, RB etc? We do not know unless we list all the outcomes.
5. List the possible outcomes
   Students may wish to list all possible outcomes directly, hopefully in a systematic way. Allow them to do this, but compare the use of grids and tree diagrams, pointing out how easy it is to miss outcomes without a diagram.
6. Use informal diagrams to list the possible outcomes
   Students may invent their own diagrams in order to find all possible outcomes. Suggestions should be listened to and compared with tree diagrams and grids.

Reflection

Whichever theoretical method or methods has been used to determine the probability of snap, this probability should now be compared with the long-run frequency that has been found experimentally. In probability it is desirable to use theoretical and experimental approaches to verify each other.Is the answer the same from both approaches? Why do you think they are different? How close does the experimental answer have to be before we accept that it is confirming our theoretical answer? There is always a finite chance that the long-run frequency might be quite different to the expected answer. All that we can do in this situation is to collect more data to obtain a longer run.
Is the game fair?
How do we know it is fair?
What is the probability of winning?
How do we know that this is the probability?

Exploring

1. Marble Snap Revisited
   Students can now work in pairs or small groups to explore a number of variations on the game. Are any of the following games fair?:

1. Bag one: 3 red marbles, 1 blue marble
   Bag two: 3 red marbles, 1 blue marble
2. Bag one: 3 red marbles, 1 blue marble
   Bag two: 1 red marble, 3 blue marbles
3. Bag one: 2 red marbles, 2 blue marbles
   Bag two: 2 red marbles, 2 blue marbles
4. Bag one: 4 red marbles
   Bag two: 1 red marble, 3 blue marbles
5. Bag one: 4 red marbles
   Bag two: 2 red marbles, 2 blue marbles
6. Bag one: 4 red marbles
   Bag two: 3 red marbles, 1 blue marble

Start with a class discussion of how we can determine whether each of the games is fair. Ensure that students are clear that they need to determine whether there are equal chances of winning and losing. Also discuss the need to use both theoretical and experimental approaches in order to verify solutions. It is a good idea to give students a format for recording their work. Sub-headings of ‘The Problem’, ‘Method’, ‘Solution’ and ‘Verification’ are useful. Students will need to record experimental results systematically – tally charts are the easiest method.

While students are using an experimental approach they can be scaffolded with questions such as:
Are you replacing the marbles after each trial?
How will you record the results?
What do you think of these results?
Do they mean that the game is fair?
What would happen if you did the experiment again?
How could you improve the experiment?

If students are using the grid method to investigate the theoretical probability they can be scaffolded with questions such as:
What is in the first bag?
Can you write those possibilities across the top of the grid?
What is in the second bag?
Can you write those possibilities down the side of the grid?
If you pull a blue marble from the first bag and a blue from the second, what is the outcome?
Where should you write this outcome on the grid?
How many different outcomes are there?
How many of these outcomes are ‘snap’?
What is the probability of ‘snap’?

If students are using the tree diagram method to investigate the theoretical probability they can be scaffolded with questions such as:
What is in the first bag?
Can you write those possibilities as the first level of a tree diagram?
What is in the second bag?
If you pulled a red marble (etc) from the first bag, what could you pull from the second bag?
Can you write those possibilities as the second level of the tree diagram?
On this diagram can you follow the branches through to list the outcomes?
How many different outcomes are there?
How many of these outcomes are ‘snap’?
What is the probability of ‘snap’?

Reflection

Hold a class discussion where students present the results from their experiment on a variation of marble snap.
Was your version fair?
How do you know?
What happened when you carried out an experiment?
Can you explain what the theoretical probability is?
Was there much difference between your experimental and theoretical approach?

2. Best Egg box
   Three half-dozen egg boxes are numbered as below.

Working in groups of three, each student is allocated an egg box. Two dice are rolled and their scores multiplied together. A counter is dropped into the hole with that number. The winner is the student whose egg box gets all the numbers covered first.
Is the game fair? Is one egg box better than the others? Is one egg box worse than the others?
The same approach used with “Marble Snap Revisited” may be used here also. Start with a class discussion of how we can determine whether the game is fair. Ensure that students are clear that they need to determine whether there are equal chances of each egg box winning. Also discuss the need to use both theoretical and experimental approaches in order to verify solutions. The same format should be used for recording their work. Sub-headings of ‘The Problem’, ‘Method’, ‘Solution’ and ‘Verification’ are useful.Students will need to record experimental results systematically – tally charts are the easiest method. Discuss the desirability to record the number of times each score is rolled, rather than just the number of times each egg box wins.

While students are using an experimental approach they can be scaffolded with questions such as:
How will you record the results?
Are you recording how often each number turns up?
What do you think of these results?
Do they mean that the game is fair?
What would happen if you did the experiment again?
How could you improve the experiment?
It may be useful to hold a class discussion and summarise the experimental results from the whole class in order to determine the long-run frequency.

If students are using the grid method to investigate the theoretical probability they can be scaffolded with questions such as:
What numbers are on the first die?
Can you write those possibilities across the top of the grid?
What numbers are on the second die?
Can you write those possibilities down the side of the grid?
If you rolled a three with the first die and a five with the second, what is the outcome?
Where should you write this outcome on the grid?
How many different outcomes are there?
How many of these outcomes are in the first egg box?
What is the probability of rolling a score that is in the first egg box?
More able students may point out that once a counter has been put in an egg box, then the probability of putting in another counter for that player will change. This is why it is useful to record how often each event occurs rather than just which player won. The number of times each event occurs may be compared directly with the experimental approach.

If students are using the tree diagram method to investigate the theoretical probability they can be scaffolded with questions such as:
What numbers are on the first die?
Can you write those possibilities as the first level of a tree diagram?
What numbers are on the second die?
If you rolled a three with the first die, what might you roll with the second die?
Can you write those possibilities as the second level of the tree diagram?
On this diagram can you follow the branches through to list the outcomes?
How many different outcomes are there?
How many of these outcomes are in the first egg box?
What is the probability of rolling a score that is in the first egg box?
More able students may point out that once a counter has been put in an egg box, then the probability of putting in another counter for that player will change. This is why it is useful to record how often each event occurs rather than just which player won. The number of times each event occurs may be compared directly with the experimental approach.

Reflection

Hold a class discussion where students present the results from their experiment on “Best Egg Box”.
Was the game fair?
Which egg box was best / worst?
How do you know?
What happened when you carried out an experiment?
Can you explain what the theoretical probability of dropping a counter in each egg box is?
Was there much difference between your experimental and theoretical approach?

3. Dice Differences
   This is a game for two players. A grid is drawn as below and a counter placed on each number.

Two dice are rolled and the smaller score subtracted from the larger. If the result is 0, 1 or 2, then player A removes a counter from that number. If the result is 3, 4 or 5, then player A removes a counter from that number.The winner is the first player to remove all their counters.
Is the game fair? If it is not, then how could you make it fair?
Again the same approach used with “Marble Snap Revisited” may be used here also. Start with a class discussion of how we can determine whether the game is fair. Ensure that students are clear that they need to determine whether there are equal chances of each player winning. Also discuss the need to use both theoretical and experimental approaches in order to verify solutions. The same format should be used for recording their work. Sub-headings of ‘The Problem’, ‘Method’, ‘Solution’ and ‘Verification’ are useful .Students will need to record experimental results systematically – tally charts are the easiest method. Discuss the desirability to record the number of times each score is rolled, rather than just the number of times a player removes a counter or wins the game.

While students are using an experimental approach they can be scaffolded with questions such as:
How will you record the results?
Are you recording how often each number turns up?
What do you think of these results?
Do they mean that the game is fair?
What would happen if you did the experiment again?
How could you improve the experiment?
It may be useful to hold a class discussion and summarise the experimental results from the whole class in order to determine the long-run frequencies for 0, 1, 2, 3, 4 and 5.

If students are using the grid method to investigate the theoretical probability they can be scaffolded with questions such as:
What numbers are on the first die?
Can you write those possibilities across the top of the grid?
What numbers are on the second die?
Can you write those possibilities down the side of the grid?
If you rolled a three with the first die and a five with the second, what is the difference in score?
Where should you write this outcome on the grid?
How many different outcomes are there?
What is the probability of getting a score of zero etc?
How many of the outcomes belong to player A?
What is the probability of player A removing a counter?
More able students may point out that once a counter has been removed, then the probability of removing another counter for that player will change. This is why it is useful to record how often each event occurs rather than just which player won.The number of times each event occurs may be compared directly with the experimental approach.

If students are using the tree diagram method to investigate the theoretical probability they can be scaffolded with questions such as:
What numbers are on the first die?
Can you write those possibilities as the first level of a tree diagram?
What numbers are on the second die?
If you rolled a three with the first die, what might you roll with the second die?
Can you write those possibilities as the second level of the tree diagram?
On this diagram can you follow the branches through to list the outcomes?
How many different outcomes are ther?
What is the probability of getting a score of zero etc?
How many of the outcomes belong to player A?
What is the probability of player A removing a counter?
More able students may point out that once a counter has been removed, then the probability of removing another counter for that player will change. This is why it is useful to record how often each event occurs rather than just which player won. The number of times each event occurs may be compared directly with the experimental approach.

Reflection

Hold a class discussion where students present the results from their experiment on “Dice Differences”.
Was the game fair?
Which player stood the best chance of winning?
How do you know?
What happened when you carried out an experiment?
Can you explain what the theoretical probability of player A / B removing a counter is?
Was there much difference between your experimental and theoretical approach?
How could we make the game fair?
There are many different ways in which the game can be altered to make it fair.
P(0) = 6/36
P(1) = 10/36
P(2) = 8/36
P(3) = 6/36
P(4) = 4/36
P(5) = 2/36
We might give player A the odd numbers, or just 1 and 2, or 0, 1 and 6 etc.

Reflecting

A final activity that may be used for summative assessment is “Cross the Stream”. This is also a game for two players and is played on a board as illustrated below. Each player is given 12 counters that they may place anywhere on their side of the stream. The players take it in turns to roll two dice and the scores are added. The player may then move one counter to the other side of the stream if they have a counter on that number. The winner is the first player to move all their counters to the other side of the stream.

Where should you place your counters to stand the best chance of winning?
If students have been recording the previous activities using the format of ‘The Problem’, ‘Method’, ‘Solution’ and ‘Verification’, then this can be used for writing up “Cross the Stream” for assessment. Reports from students would be very suitable for inclusion in portfolios or as posters.
The activity can also be used for teaching, of course, in which case the format of the previous activities may be followed.
Start with a class discussion of how we can determine which the best numbers to place counters on are. Ensure that students are clear that they need to determine what the probability of rolling each score is. Also discuss the need to use both theoretical and experimental approaches in order to verify solutions. Students will need to record experimental results systematically – tally charts are the easiest method.

While students are using an experimental approach they can be scaffolded with questions such as:
How will you record the results?
Are you recording how often each number turns up?
What do you think of these results?
Which are the best numbers to place your counters on?
What would happen if you did the experiment again?
How could you improve the experiment?
It may be useful to hold a class discussion and summarise the experimental results from the whole class in order to determine the long-run frequencies for 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12. Students should very quickly realise that a score of 1 is impossible and that some scores are much more likely than others.

If students are using the grid method to investigate the theoretical probability they can be scaffolded with questions such as:
What numbers are on the first die?
Can you write those possibilities across the top of the grid?
What numbers are on the second die?
Can you write those possibilities down the side of the grid?
If you rolled a three with the first die and a five with the second, what is the total score?
Where should you write this outcome on the grid?
How many different outcomes are there?
What is the probability of getting a score of two etc?

If students are using the tree diagram method to investigate the theoretical probability they can be scaffolded with questions such as:
What numbers are on the first die?
Can you write those possibilities as the first level of a tree diagram?
What numbers are on the second die?
If you rolled a three with the first die, what might you roll with the second die?
Can you write those possibilities as the second level of the tree diagram?
On this diagram can you follow the branches through to list the outcomes?
How many different outcomes are there?
What is the probability of getting a score of two etc?

Reflection

Hold a class discussion where students present the results from their experiment on “Cross the Stream”.
Which was the best number to place a counter on?
Which was the worst?
How do you know?
What happened when you carried out an experiment?
Can you explain what the theoretical probability of moving a counter placed on two, seven, five etc?
Was there much difference between your experimental and theoretical approach?
What do you think the best arrangement of counters is?
Clearly one counter on every number is not the best arrangement. Neither is all the counters placed on seven.
P(1) = 0/36
P(2) = 1/36
P(3) = 2/36
P(4) = 3/36
P(5) = 4/36
P(6) = 5/36
P(7) = 6/36
P(8) = 5/36
P(9) = 4/36
P(10) = 3/36
P(11) = 2/36
P(12) = 1/36
The best arrangement is when the number of counters on each number is proportional to the probability of rolling that score.
This gives:

1. 0
2. 1/3
3. 2/3
4. 1
5. 4/3
6. 5/3
7. 2
8. 5/3
9. 4/3
10. 1
11. 2/3
12. 1/3

Of course we can only place whole numbers of counters. This means there is room for discussion about where they should be placed.

Students may of course dispute this, in which case this arrangement may be played against the one that they prefer a large number of times. I usually bet a chocolate fish that I am right!