Why didn't the Athenian army go out to fight the Persians when they arrived at Marathon?

Answer 1

They were in the middle of a religious festival but undertook to come in a couple of days when it was over. They then set off at a fast pace, but before they arrived the battle was fought. They marched up to Marathon, viewed the battlefield, congratulated the Athenians and went home.

Answer 2

They were waiting for the reinforcement of the Spartans, who had a religious festival going on and wouldn't arrive for 10 days to help them. In the interim the Athenians lurked in the hills above the Marathon plain, where the Persian cavalry couldn't get at them.

On the tenth day they observed the Persian cavalry being embanked onto the Persian ships and seized the opportunity to rush down and defeat the inferior Persian infantry unprotected without its cavalry. The Persian cavalry were shipped around to Athens where traitors were set to open the city gates to let them in. However the Athenian infantry realised what was happening and ran back the 26 miles over the hills from Marathon and formed up in front of the city just as the Persian cavalry was disembarked. Frustrated, the Persians went home.

This run back is the origin of today's Marathon race, the difference being that the Athenians were in sandals and carrying their armour and weapons and had already just fought a battle - something which our pampered runners of today might find a little difficult.