Even though the supposed 'answer' has been posted there were many errors in your post which would lead to many people answering wrong.
Firstly you did not specify the state or even country, in some places as you mentioned later emergency vehicles do have the right of way in emergencies, under no circumstances is a mailman one who responds to an emergency, even though he is federal he is not above state laws and again, most states have laws saying he has to yield.
Also this was a tricky question, and in many tricky questions the trick is you have to take it literally, and in the literal sense we don't know how many vehicles were even at the intersection. You said there was a cop, a mailman, an ambulance, and a firetruck. The cop could have been driving the firetruck and the mailman could have been driving the ambulance, therefore there could have been only 2 vehicles. Sounds rediculous but a lot of trick questions rely on people assuming things that seem to make the most sense (that the cop is in a police car and the mailman is in a mail truck)
Before I read your reply I was actually going to post a theory involving just two vehicles because again, many trick questions rely on that to fool people.