median
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)

thanks to Jo Morgan, my slides are available in a ppt version that can be found here

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