[sdab]

Yep, its a nice problem. Reformulating it such that there are two possibilites (the coins are either different or the same) rather than four (permutations of two coins) is the way to solve the problem.

I was going to post a hint rather than the solution, but you beat me to the punch.

[placeybordeaux]

I liked showing the four possibilities and thinking about it in the way that one of them has to be responsible for winning in each case.

[SMackinnon]

Yeah, added a little spoiler tag.