don steward
mathematics teaching 10 ~ 16

Monday, 28 March 2016

creating a fair game

this fascinating probability task has been described in detail by Dr Natalya Vinogradova at Plymouth State University (New Hampshire, USA)

there is a pdf version of her article

thanks to Jo Morgan, my slides are available in a ppt version that can be found here
(download it for the animations)

Natalya suggests that it is helpful for students to initially undertake a practical version of the task

maybe comparing the results for different number pairings (with 6 altogether)

an analysis of these results, for three of each, could involve a simple 'tree' diagram

from this diagram, are you more likely to obtain two of the same colour or two different colours?
what makes you say that?

a more normal tree diagram shows the unfairness of this game
what happens for other colour splits?
more likely to win with a 5 : 1 colour split

certain to win if they are all the same colour...

a 4: 2 colour split is the fairest option
for any even number of counters with an equal split of colours, a proof that you are more likely to obtain two different colours than two colours that are the same:

try a different number of counters to see if a fair game can be created
this works!

so, it seems, do other pairs of consecutive triangular numbers

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