So, in D&D's 5th edition, when you hit zero hit points, unless you've taken enough such that the remaining damage you take in that hit would equal your maximum HP, or if an effect explicitly says that a creature reduced to zero hit points by it automatically dies, you instead go unconscious and begin rolling death saves on your turn.
This is basically the "bleeding out" mechanic - you're not dead, but you're in serious danger if someone can't patch you up ASAP. In practical terms, it gives the party a bit of a grace period to ensure the party member survives. Depending on the tone and difficulty of your campaign, a DM might make monsters opportunistic - seeing a chance to truly kill an opponent (which is a lot tougher on the players) versus moving on to conscious targets (and just so happening to make it more likely the party can heal their friend.) A monster/foe can choose to attack the unconscious target, taking advantage of the fact that melee attacks are automatic crits to quickly give the unconscious character three failed death saves (as any damage gives one and a critical hit gives two,) though my tendency is to avoid doing that in early levels when the players have little recourse, unless I wind up running a grim and gritty-style campaign where we're expecting heroes to die easily - instead I have the foes decide that they can always come back to this target once the others are down (and, frankly, there is something to be said for monsters that fight to survive, rather than to kill the party).
Anyway, the rules on rolling death saves are fairly simple - on your turn, you roll a d20, and if the result is 10 or higher, you get a success, while 9 or lower is a failure (this gives you a slight advantage - 55% chance at success and 45% at failure). This is actually the only other type of roll where, rules as written, rolling a 1 or a 20 has a different result, the other being attack rolls. Here, a 1 means two failures, but a 20 means not just success, but that you come back to consciousness with 1 hit point, meaning you are now conscious, all your death saves reset to 0, and you can actually do things (which hopefully involve healing yourself).
What's interesting, though, is that if you have 3 successes that aren't 20s, you merely stabilize - you're still unconscious, but you are no longer rolling death saves and, as long as you don't take any more damage, you'll wake up in an hour with 1 hit point.
I'll keep the particular show unsaid for spoiler reasons, but in a streamed D&D show, during the climactic fight against the big bad of the campaign, the cleric in the group had gone unconscious. Going for three rounds and rolling all successes, they stabilized. But when the fourth round came (as the group's primary healer, there weren't many who could help them) the player asked if they could roll another death save. I can't recall if the DM said this would mean they were opting into once again being unstable, and thus in danger of dying, but they allowed it. The next roll was a natural 20, which allowed the cleric to get up, heal the rest of the party, and ultimately turn what was getting close to a TPK into a win.
That made me think:
From a game point of view, it makes sense that you'd have to risk something to get the reward of potentially rolling a natural 20 and regaining consciousness. And DMs of course don't need to kill you if you're stable and unconscious - you can always take the players prisoner if you don't want to see the campaign collapse. Still, it makes sense in a game design sort of way that, hey, if you want that 5% chance to spring back into the fight, you're going to need to risk the... I don't have the patience to work out the probability you'll die percent chance, though I think it's bigger than 5% (though the most likely thing is, I'd guess, that you eventually stabilize again).
But from a logical standpoint: doesn't it make sense that if you could jolt back awake in the midst of bleeding out, you could also jolt back awake after your blood has coagulated enough to seal up the wounds? Indeed, wouldn't it be more likely?
So, I could imagine a system in which you simply keep rolling but can no longer fail the saves.
That being said, one of the things that D&D sometimes struggles with is the way that once initiative ends, the precision by which time is divided becomes harder to manage. If you were rolling every six seconds, the chance you'd never roll a natural 20 in the 600 rounds that represent an hour is 43 in a quadrillion (if my math is correct - basically it's 0.95 to the 600th power). So, you know, it'd be pretty rare to go that long without rolling a 20.
So, maybe the rule works out better this way.
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