So before I get to the steps to solving, we should first define what a fair game is. A fair game is a game where there is equal chance of win or loss, so for this to be a fair game, the expected value must be zero.
Now that we defined what a fair game is, let's get to solving! Firstly, we want to get all the possibilities and multiply their outcomes with their chance. In this case the possibilities are {1,2,3,4,5,6}, all have a 1/6 chance, and {1} outcome is 0, {2,3,5} outcome is 4 and {4,6} outcome is -6:
[tex]\textsf{1:}\ 0\times \frac{1}{6}=0\\\\\textsf{2:}\ 4\times \frac{1}{6}=\frac{4}{6}\\\\\textsf{3:}\ 4\times \frac{1}{6}=\frac{4}{6}\\\\\textsf{4:}\ -6\times \frac{1}{6}=-\frac{6}{6}\\\\\textsf{5:}\ 4\times \frac{1}{6}=\frac{4}{6}\\\\\textsf{6:}\ -6\times \frac{1}{6}=-\frac{6}{6}[/tex]
Next, take all the products and add them up:
[tex]\frac{0}{6}+\frac{4}{6}+\frac{4}{6}-\frac{6}{6}+\frac{4}{6}-\frac{6}{6}=\frac{0}{6}=0[/tex]
Since the expected value is zero, it is true that this is a fair game.
