I'm not a fan of this approach at all. I don't want the success of the party to depend on a lucky guess. I prefer an approach where the players have things to consider about the entrance. I use this approach for intersections in the dungeon too. Sure, there are always unknowns. The players don't know the layout of the dungeon after all. But I prefer that they make an informed choice. For example:
-One point of entry may involve a heavy stone door. It may cause a lot of noise when opened, or it may require a strength check to get it to open, but it also offers a direct way in.
-Another point of entry may require the players to swim underwater, which is kinda scary, but ultimately may give them the element of surprise.
-Another point of entry may require the players to sneak past a powerful foe. Or through a tunnel covered in cobwebs.
-Maybe one of the doors has a scary demon face on it?
-Maybe one of the doors has a complicated lock?
To me this is far more interesting than three identical looking entrances.
Fair enough.
My question, to follow on from your example above, then becomes: how much of this info do the PCs have before making their decision? Do they know the heavy door will make noise, or is that just speculation on their part? Do they know there's a powerful foe waiting in the other entrance, without taking steps to find out? Etc.
Because if not, then it still comes down to trial by guess; only much more dressed-up. Do we enter via:
Heavy stone door
Underwater passage (if it even goes where we want it to go!)
Passage that looks clear
Passage that looks cobwebby
Scary demon-face door
Imposing-lock door
This is a situation where in my view it's absolutely vital the DM not give out any info the PCs either can't know about or haven't earned by observation/scouting. (think about if this were real, how much would people be able to discern on approaching and-or observing such a structure, and give out information based on that)
If they start using skills such as tracking, or spells of divinaton magic e.g.
Detect xxxx, those can give them more info if there's any to give. For example scary-demon-face door might radiate magic because the demon face is a permanent illusion; tracks could be found going in-out of the clear passage indicating that it's occupied, and so forth. The underlying idea here is to make them earn this information rather than giving it out for free, with full acceptance that doing it this way will slow things down.
On a more macro level, I've always seen luck as being a big - if not the biggest - overall factor in the game as a whole. If it wasn't, we wouldn't use dice.
