The third-age mage hat is a member's only piece of armour that is part of the Third-age mage set. To wear the mage hat the player needs at least 65 Defence and 65 Magic. To obtain a Third-age mage hat the player can either receive one from a level 3 Treasure trail, buy it on the GE, or buy from another player.
It was long thought that the chances of obtaining a piece of third-age armour was of about 1 in 10,000 (according to Paul Gower), and in 2010 Mod Maylea stated that the odds are lower than 1 in 100,000 . However, a livestream with Mod Simon and James revealed the mechanics of Treasure Trail rewards in detail, meaning the actual chances could be calculated: approximately 1 in 16,896 per item roll for any of the 12 parts of third-age equipment, or approximately 1 in 202,752 for a specific part, per item roll - with an average of 5 item rewards per Trail, this gives approximately 1 in 3,380 for any part, or 1 in 40,551 for a specific part, per Trail completed. For a more detailed breakdown of drop mechanics and calculations, see Treasure Trails/Rewards#Hard.