I don't have a problem with the 78% figure (with the caveat that the exact number is a little arbitrary), but there is a huge difference between a 78% chance to hit and a 78% chance of success in an encounter.
Sure.
An encounter is made up of a large number of rolls. If "victory" in the encounter goes to the party that gets more successes (a vast oversimplification, but bear with me), the probability of success in an encounter will be vastly higher than 78%, if the players hit 78% of the time.
Ah, it's your oversimplification that is the problem. Victory in the encounter doesn't go to the team that gets the most successes, it's vastly more complex than that. (They can get fewer successes, but do more damage; they can get fewer successes but concentrate them on an enemy 'leader', causing the rest to flee; they could get many more successes, with each having less impact; and so on.)
Basically, I would advocate setting enemy defences to a level where the PCs can hit on a 5 or 6, and then upping the monster numbers or hit points so that the chance to win the encounter is about right. (Or, conversely, increase the monster attack bonuses and/or damage per attack, but that tends to make for a more 'swingy' combat, which may not be desirable.)
Also, 78% seems like a plausible "gut-check" number when it comes to succeeding on a combat round, but it seems crazy low on an encounter level. If an average adventure is a 10 encounter (i.e. one level) story or dungeon, then you suggest that the players fail 2-3 of those encounters per story.
On the first attempt. Bear in mind that the PCs should usually be able to make multiple attempts at a given encounter - the options shouldn't be "win or die"!
(It's probably also true that the 78% figure should only hold if the PCs go at the encounter 'cold' - that is, they dive in and start rolling dice. If they approach with a bit more caution, work out a plan ahead of time, and basically "play smart", then the success rate should go up accordingly.)
That's really high! I'm sure there's a fun RPG with that dynamic, but I'm know that most D&D PCs don't expect to lose approximately 1 in 4 combat encounters.
You are right about this. The common expectation these days seems to be that PCs will win through on 99% of encounters on the first attempt, and the remaining 1% will end in a TPK. I don't think that's a good thing; in general, I think the game is actually more satisfying (for all involved) if the players
don't have things their own way - they have defeats and set-backs, but they do eventually win through by perseverence and skill.
(It's also perhaps worth noting that this isn't exactly alien to video-game players - they generally don't have a problem taking multiple shots at a section of the game, and especially "boss" encounters. Of course, video-games have the easy option of loading a saved game, which a traditional RPG lacks, but the concept is the same.)
Oh, and you're right - the 78% figure is quite arbitrary. It does, however, feel "about right".