Solution:
There can be two cases.
Probability of X losing is (100 - 50)% or 50% and probability of Y losing is (100 - 40)% or 60%.
Case (1) X wins and Y loses. This probability will be 50% *60% = 30% .
Case (2) Y wins and X loses. This probability will be 40% * 50% = 20% .
Or required probability is 30% + 20% = 50%

gmatusa2010 wrote:could this problem be flawed?

Deeply, deeply flawed. It's open to different interpretations, for one thing, and the solution that you pasted in from the source is wrong, at least if there can be only one winner.

While it isn't at all clear what the question means, I imagine they're thinking about a game that one player plays, then the other player plays; the two people are not playing against each other. So perhaps they're each doing something like playing chess against a computer, and X is a bit better than Y. Now, when the question says that 'only one can win', I can only guess that it's just trying to make clear that it's asking for the probability of exactly one of them winning (that's why their solution is wrong: they are including the possibility that X and Y both win). I don't think they're trying to say that it's impossible that both players win. Still, I interpreted it the same way as you did the first few times I read it, so the wording is terrible here. Indeed, if it is impossible for both people to win, then the probabilities aren't independent, so the question becomes mathematical nonsense.

Questions in Probability/Counting tend to require more careful wording than questions in any other GMAT area. Sources that can't be bothered to word their probability/counting questions precisely are going to be more confusing than helpful, and I'd recommend you not waste your time with them. Where is this question from?