Max and Millie Dice Game

Question

Max and Millie have invented a dice game. They roll two four-sided dice and look at the difference between the two numbers that come up. Max scores a point if the difference is 1 or 2. Millie scores a point if the difference is 0 or 3

 

(a) is the game fair? give calculations and reasons to support your answers

 

(b) how could you change the game to make it more fair?

Answer 1

There are 4 possible numbers to roll with the first die and 4 possible numbers to roll with the second die. That's a total of 4×4 = 16 possible outcomes.

 

Those outcomes are:

 

1,1 - Millie's point (0)

1,2 - Max's point (1)

1,3 - Max's point (2)

1,4 - Millie's point (3)

 

2,1 - Max's point (1)

2,2 - Millie's point (0)

2,3 - Max's point (1)

2,4 - Max's point (2)

 

3,1 - Max's point (2)

3,2 - Max's point (1)

3,3 - Millie's point (0)

3,4 - Max's point (1)

 

4,1 - Millie's point (3)

4,2 - Max's point (2)

4,3 - Max's point (1)

4,4 - Millie's point (0)

 

Notice that even though 1,2 and 2,1 have the same difference, we need to list both of them. That's because the 2-1 outcome will come up twice as often as the 1-1 outcome.

 

 

 

So, that's a total of 6 of the 16 outcomes that give Millie a point (four 0's and two 3's) and 10 of the 16 outcomes that give Max a point (six 1's and four 2's).

 

The game is not fair, since Max has a 10/16 chance of getting a point while Millie only has a 6/16 chance.

 

 

 

You can make the game fair by giving Max a point for a 0 or 2 (4/16 + 4/16 = 8/16 chance) and giving Millie a point for a 1 or 3 (6/16 + 2/16 = 8/16 chance).

Answer 2

Rolling the dice we are only looking at the two numbers rolled

Not the order in which they are rolled

So rolling a 1,2 is the same as rolling a 2,1

 

The only possible outcomes then are

 

1,1   difference = 0 = 1  point Millie

1,2   difference = 1 = 1point Max

1,3   difference = 2 =  1point Max

1,4   difference = 3 = 1 point Millie

 

2,2   difference = 0 = 1 point Millie

2,3   difference = 1 = 1 point Max

2,4   difference = 2 = 1 point Max

 

3,3   difference = 0 = 1 point Millie

3,4   difference = 1 = 1 point Max

 

4,4   difference = 0 = 1 point Millie

 

Thus they each have 5 chances to get a point and the game is fair