Actually, both 48% and 84% are correct answers. Why?
Because there's something important .Roy didn't say: Do you proceed to try to get such "something" even if you do get it in the first try?
If you get the thing in the first try and you proceed, then the chance that you get the thing in exactly one try is 48% (0.6*0.4+0.4*0.6).
If you get the thing in the first try and you DON'T proceed to try again, then the chance is 84% (0.6+0.4*0.6)
It's my two cents.
I'm too lazy to draw the tree diagram...anyone?
Edit: My assumption is that: The event that you get the thing is independent i.e. the chance you get the thing is always 60%. I'm not talking about the case that there are 10 things, 6 of which are what you want (where you have 60% chance to get some thing you want)... just like what John says in the next post.