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Random Squares

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

 

These grids of Red and Yellow small squares are generated randomly; each small square has the same probability of being red.

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Clearly we can describe the grids completely by specifying the colour of each and every small square. Are there any other ways that you might describe the patterns in less detail but still giving a flavour of the pattern?

Does any order emerge out of the randomness?

  • Can you estimate the distribution of the number of coloured regions that will occur in a randomly filled grid?

Can you somehow 'categorise' certain patterns?

  • How many ways of 'classifying' the patterns into groups can you think of?
  • Are certain categories of pattern more or less likely than others?

Various other probabilistic concepts can be explored using this tool:

  • Can you estimate the probability of two opposite sides of a grid being joined by a solid line of one colour?
  • If someone said that they had used the tool to create a chequered configuration like a chess board, for what sized grids and probability of reds would you believe them?



NOTES AND BACKGROUND

As the saying goes, there are lies, damned lies and statistics. One important use of statistics is to decide if a set of data (such as a tax return) is suspicious in any way. Whilst all grid configurations created by this tool are equally likely, certain grid configurations (such as entirely yellow or those which are highly symmetric) might raise suspicions that the grid was not generated randomly. Statistics is in many ways an art and engaging with this problem will help develop a feel for randomness.

The problem also leads into many of the ideas surrounding information theory. In essence, the more information required to describe a system exactly, the more 'random' it is. These ideas are of great importance today in communication and networking.