If you cast a spell with flash during an opponent's turn, do you still have to pay its full mana cost?

Flash (CR 702.8) grants a single, narrow permission: it allows you to cast a sorcery-speed spell at any time you could cast an instant — including during an opponent's turn. That is the only thing flash does.

The mana cost of a spell is determined entirely by what is printed in its upper-right corner (or defined by its rules text). CR 202.1 states that a card's mana cost is set on the card itself. CR 601.2f requires you to pay all costs when casting a spell, and flash provides no cost reduction whatsoever.

Cost reductions come from completely separate effects — such as cards with explicit cost-reduction text, the Convoke or Delve mechanics, or effects like Goblin Warchief. None of those are tied to flash.

Concrete example: Your opponent attacks on their turn and passes to combat. You control no blockers, but you have a Briarbridge Tracker (a creature with Flash) in hand and enough mana. You may cast it during their combat phase because of Flash, but you must still pay its full printed mana cost of {2}{G}. Flash got you the timing window — nothing else changed.

In short: Flash = timing permission only. Always pay the full mana cost unless a separate effect specifically reduces it (CR 702.8a, CR 601.2f).