Same Or Different

Task 18 ... Years 4 - 12

Summary

This is a game for two players. They place a number of blocks (two colours only) into a bag, then close their eyes and choose one block each. In advance it has been decided that one player will get a point if the chosen blocks are the same colour and the other gets a point if the blocks are a different colour.

Clearly there would be an unfair advantage to one player for some combinations of colours in the bag, eg: (6,1). The problem is:

Which combinations of colours produce a fair game?
and students are told there are two fair games using any mixes of colours up to (6,6) 		 

Materials

  • Twelve cubes - 6 in each of two colours
  • A bag or other container for hiding the blocks

Content

  • chance - simple probability
  • long run frequency
  • conditional probability
  • concept of 'fairness'
  • statistical inference from data
  • sample size and intuitive levels of confidence
  • recognition of all possible outcomes (sample space)
  • generalisation - recognition of triangle number patterns
  • combinations function
  • complementary probability
    Pr (A') = 1 - Pr(A)
Same or Different

Iceberg

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

  • How do we decide if a game is fair?

    If by experiment we would expect about an equal number of points for each player, but how many experiments is enough to decide this 'equality'?

    If by symbolic reasoning, eg: for the (3,1) game, what equations can be written for other combinations.

    Let (3,1) stand for a bag with 3 Red and 1 Blue, then
    Pr (SAME) = Pr (RR) + Pr (BB)
    = 3/4 x 2/3 + 1/4 x 0/3
    3 ways to choose the first red block from the 4 blocks in the bag followed by 2 ways to select the second red block from the remaining 3 blocks
    +
    1 way to choose the first blue block from the 4 blocks in the bag followed by 0 ways to choose the second blue block from the remaining 3 blocks because they would all be red.
    = 6/12
    = 1/2

    Pr (DIFFERENT) = Pr (RB) + Pr (BR)
    = 3/4 x 1/3 + 1/4 x 3/3
    = 3/12 + 3/12
    = 6/12
    = 1/2
    So, (3,1) is a fair game and analysing other likely combinations in the same way reveals (6,3) is the other fair game up to (6,6).

  • If there are two fair games using blocks up to (6,6) are there more if we use more blocks?
  • List the combinations which make fair games. Is there any pattern in this data? Can it be used to predict other fair games?

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. 		   

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 153, Same Or Different, which also extends the investigation with companion software. You will need multiple sets of blocks and one bag or container per group.

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 Same Or Different task is an integral part of:

  • MWA Chance & Measurement Years 3 & 4
  • MWA Chance & Measurement Years 9 & 10

The Same Or Different lesson is an integral part of:

  • MWA Chance & Measurement Years 9 & 10

Green Line

From The Classroom

Damian Howison, MacKillop College Swan Hill, reports on a 'tweak' to this investigation that developed when using it at the top level of Secondary School. The images were exported directly from the classroom Interactive White Board and suggest how the lesson developed.
I used the task in my Year 12 Methods class last week and we had a good time with it. In particular we used simple line diagrams to show the probabilities and this naturally lead to using the Combinations Function for calculating the probabilities (incidentally we derived the combinations function last year using Task 129, Farmyard Friends).

Using the combinations approach made it straightforward to search for fair games on Excel using COMBIN. The pattern 1, 3, 6, 10 came off the screen and one student recognised these numbers as 2C2, 3C2, 4C2, 5C2...

This suggested that one solution for a fair game was the general case of ( NC2, (N+1)C2 ).

And because of the combinations relationship with Pascal's Triangle, this sequence is also one of the diagonals of the triangle!

So as you can imagine, it was a wonderful little lesson of discovery - the best type where I the teacher learn just as much as the students.
Damian has also given us his class's spreadsheet to be freely shared with all users of Same or Different.
Spreadsheet: Year 12 Mathematics Methods (Left click to view. Right click to save.)

Another Thought

Download Damian's sheet and try it. At the bottom right end of the rows and columns it looks like this:

It seems there are two ways to find fair games:

  1. Sequential triangle numbers (TN, TN+ 1) that give exactly a probability of 0·5 for selecting either same or different for any value of N.
  2. (N, ~N) where ~N = values close to N.
    As N gets larger the probabilities approach 0·5 as seen by the main diagonal in the spreadsheet:
    0·333, 0·400, 0·429, ... 0·485, ... 0·490, ... 0·498, ... 0·499 ...
    suggesting that the limit as N approaches infinity of:
    2 x (NC2) / (2N)C2 = 0·5
    (see Slide 3 above).
There's a lot more Year 12 work lurking here than might be apparent at first glance.

Green Line
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