Secondary Maths - introducing probability

Asking a question ‘What is the probability of rolling two fair dice and obtaining two sixes’ can generate all kinds of misconceptions and incorrect answers;  this question reflects the underlying challenges associated with teaching probability theory effectively.
I first observed the brilliance of ZILCH attending a Masterclass for Gifted and Talented pupils based at Bath University.    A unique feature of the game is the ability to generate interest and enthusiasm across the age and attainment range;  a top set year 7 group can be just as engaged as a foundation year 10 group.  The game is played using packs of six dice and ideally the game is played in pairs.  A scoring system is clearly explained as follows:

Scoring 3 Two’s all in one go  =   200 points
Scoring 3 Three’s all in one go = 300 points
Scoring 3 Four’s all in one go =  400 points
Scoring 3 Five’s all in one go  =  500 points
Scoring 3 Sixes all in one go = 600 points
Scoring 3 One’s all in one go = 1000 points
Scoring  1,2,3,4,5,6  all in one go  =  1500 points  
Scoring 1 One  =  100 points
Scoring 1 Five = 50 points

Each player rolls one dice and the highest score takes the first turn.  All six dice are rolled and the scoring dice are placed to one side.  If there are no scoring opportunities consistent with the above scoreboard the player scores nothing – ZILCH – and it is the turn of the next player.  If some of the dice are scoring dice, the player uses intuition and skill to determine whether to ‘stick’ or to ‘gamble’ to improve their score.   For example, the first roll of the dice might yield: 4, 4, 4, 4, 2, and 6

Three fours together score 400 and the player decides to roll the three non scoring dice again with the hope of improving a score of 400 points.   The 4, 2 and 6 are picked up and rolled again in the hope of scoring three of the same number or at least rolling one five or one 1.  If nothing is scored the gamble has not paid off and the player ‘ZILCHES’ thereby losing their score for that turn.
The first player to score 2000 (or more) points is the winner.  

Following a few games of Zilch, the introduction of a Zilch World Cup has generated great interest and it is intriguing to witness that ‘lady luck’ seems to follow certain individuals around, especially in a World Cup Final.  Introducing the ‘Zilch Dilemma’ further enhances the conceptual understanding of probability;  if you have a good score of say 350 points in your first throw, is it best to ‘gamble’ or to ‘stick’ with two dice remaining?  Representing the possibilities on a 6 by 6 sample space diagram illustrates that the chances of a gamble increasing a score is higher than we may intuitively think – 20/36 or 5/9.   The sample space diagram also makes our initial question crystal clear to students;  the probability of rolling two fair dice and both dice landing on a 6 is 1 chance in 36.   Gifted and Talented groups may wish to find the probability of scoring 1,2,3,4,5 and 6 in one throw or the probability of winning in their first throw with an extraordinary throw of six one’s!

Chris Curtis
Curriculum Team Leader for Mathematics