The easiest way to see this, and the solution usually given is that if you choose a door and not switch, the probability you were right is 1/3 -- there's one treasure door and 3 doors in total.
If you chose a door, then switched, you would get the treasure if the one you switched from was a dragon (ie you switched from dragon to treasure). There's two dragon doors and 3 doors in total, so this probability is 2/3.
Of course you can do it using conditional probabilities, but that would take more work than is required for this problem
