# Fair or not fair

Mike Rath
21st November 2016 at 15:02

In the M series of QI, one programme was about maths.  One interesting fact that came up in the programme was about what happened when you tossed two coins.  MPs were apparently asked about the probability of getting two heads – and something like 60% of them got the answer wrong!!  This gave me some idea of the general knowledge about probability.  It can often be great fun to look at this particular topic in school.

Volunteering in my local primary school, I am often given small groups of bright students to extend.  Recently, I took a group of 5 Year 2 students and decided to play some dice games with them.  I started with my usual game.  We discussed the possible outcomes of rolling two dice – and came up with the fact that you could score anything from 2 to 12.  I asked them to choose their outcome of choice and we started rolling the dice.  If their number came up, they scored a point – no matter who rolled the dice.

When we counted up their scores, I began a discussion as to whether the game was fair.  As part of that discussion, I asked them to fill out a table for me.  It was a simple adding table, with 1 – 6 along the top and 1 – 6 down the side.

 2 3 4 5 6 7 3 4 5 6 7 8 4 5 6 7 8 9 5 6 7 8 9 10 6 7 8 9 10 11 7 8 9 10 11 12

This is what the completed table looked like and, of course, a discussion about it made it clear that some numbers appear more than others, suggesting our game had not been fair.

To try to highlight its unfairness, for the first time when doing this I tried something a little different.  I realised that, with 5 students plus me, I could actually arrange a fair game instead of an unfair one.  There are 36 squares in the table above and with 6 people involved – including me – if each person had access to 6 squares, we would each have a fair chance of winning.

Therefore, the student choosing 7 would only have one number, but with a choice of any other number, I could offer a second choice to make it fair.  So, for example, the student choosing 6 would also be offered 12, so a point was scored if either number came up as the total of the two dice.  A student choosing 10 was also given 4.

This game actually turned out to be quite significant.  The students had noted in the first game that there were very different outcomes for different numbers – but they all agreed that in the second game all totals were much closer together.  We then could have an interesting discussion about fair games and unfair games.