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Your odds are about as good as they are of hitting a slot machine jackpot on the trip, I’m afraid. While you’ll have priority over complimentary upgrades (except for Global Services), there’s a good chance nobody will be upgraded on that flight if they’ve already sold half the cabin. Then, throw in the fact that there are a lot of expiring certificates, and it’s likely you won’t be the only one trying to upgrade. Finally, UA will sell upgrades to passengers at check-in, and it’s a long enough flight that people will bite (especially if they did hit the aforementioned jackpot).

Your best bet is that a couple of people oversleep and miss their flight, leading to you getting upgraded when they no-show.

Good luck!