Incidentally, there is an overall 59.5% chance of stabilizing at all with the standard death save rules and assuming no external interference nor bonuses to the save. That is including odds of getting a natural 20 on any of the first four rolls.

Round up to 60%, and you can save yourself rolling the odds out for any given NPC you care enough to check for after combat is over by rolling a d20 and saying he died if he gets an 8 or less.