## Games and Probability – Combinations of 3 Dice

To start a probability unit of work with year 9 top set, I decided to use a nuffield resource called Three Dice.

The project is based around a game in which the students draw a 3-by-3 grid in thier books and pick nine numbers between 3 and 18 (the possible total score from rolling 3 dice) to go anywhere in the grid without repeating numbers. E.g.

You can use this billiant pdf applet to roll the three dice so that the students can cross off numbers in their grid when a score comes up. The person first in the class to cross off all the numbers is the winner.

I introduced the lesson by starting with this game and purposefully did not tell them to be on the look out for anything – it was just a fun game to play from their persepective. When certain numbers like 9, 10 repeatedly came up the students said that there was something wrong with the applet – they said that it was biased!

Of course, there are 25  different ways to score 9 (E.g. 6 – 2 – 1, 2 – 1 – 6, 3 – 3 – 3, etc) with three dice and only one way to score 3 (1-1-1) or 18 (6-6-6)  so the applet was not at all biased.

I posed a problem for the students to think about:

If the applet is not biased which numbers would you put into the grid, with no repetitions, to maximise your chances of winning the game ?

I posed this without really going into anymore detail because I wanted to see if any groups would approach the problem from an interesting angle. It took some of them a good 10 minutes to understand what was going on and many groups at this point started working out all the possible combinations for the different scores (I plucked this list from the internet if you´re interested in knowing all of them).

I moved around the room asking them how they would know if they accidentally missed any of the combinations out. Many students at this point were able to tell me that there has to be 216 possible combinations but strangely enough they struggled to explain why this was the case.

After some proding with a sample space diagram for 2 dice, students started to understand that for 3 dice it would be a sample space cube with dimensions 6 by 6 by 6 and were therefore able to explain why there would have to be 216 possible combinations.

When the students started to draw up the lists, some of them noticed the symmetry of the problem. E.g. that getting a 3 had the same number of combinations as getting 18, getting a 4 had the same as 17, etc. Some of them also noticed a pattern for the total number of combinations for each score, namely the triangle number sequence. For example,

A score of 3 –  1 combination (1-1-1)

A score of 4 – 3 combinations (1-1-2, 1-2-1, 2-1-1)

A score of 5 – 6 combinations (1-2-2, 2-1-2, 2-2-1, 3-1-1, 1-1-3, 1-3-1)

A score of 6 – 10 combinations etc.

I love the fact that this sequence continues up to a score of 8 but it does not work after that. For example, it goes 1, 3, 6, 10, 15, 21, 25, 27, etc. In the second lesson, we used this PowerPoint to analyse the problem in greater detail and find out why the triangle number sequence appears and why it stops working after a score of 8. We also discussed the idea of relative frequency (experimental probability) vs theoretical probability.