I don't know how to do this
BUT
Well the probablility of getting double 2 or double 6 is 2/36 = 1/18
So he has 1/18 chance of winning $15
and a 17/18 chance of losing $5
So maybe
$$\left({\frac{{\mathtt{1}}}{{\mathtt{18}}}}{\mathtt{\,\times\,}}{\mathtt{15}}\right){\mathtt{\,-\,}}\left({\frac{{\mathtt{17}}}{{\mathtt{18}}}}{\mathtt{\,\times\,}}{\mathtt{5}}\right) = {\mathtt{\,-\,}}{\frac{{\mathtt{35}}}{{\mathtt{9}}}} = -{\mathtt{3.888\: \!888\: \!888\: \!888\: \!888\: \!9}}$$
So the person should expect to lose $3.89
This probably is not correct. I am totally guessing. Sorry.
If you find out how it is done please let us know. If I find out I will post it on this thread. 
Someone else probably knows how to do it. I shall ask around. :))