What are the odds of a person becoming undead if they are not killed by an undead creature such as a vampire, nor animated intentionally by a spell caster?
Pretty low.
Does a proper burial help prevent this?
Proper burial prevents it 100% of the time. If you put a body in properly consecrated ground ('Hallow') and or use the appropriate divine invocation the body ('Lay to Rest') will never become undead spontaneously and cannot be animated by evil magic. In fact, this is one of the major duties of the clergy in most towns.
Unfortunately, if you are using invocations to keep bodies and spirits at rest, then you are losing the opportunity to heal wounds, cure disease, and place blessings on the living. In the event of catastrophe, something has to give.
Thus most large towns have emergency measures designed to contain the problem. Maze-like catacombs and sealed vaults beyond areas of consecrated ground and wards against the undead can ensure that even if you do get an undead problem, it will be contained. If mass burials are required that overwhelm the ability of the clergy to deal with the problem, physical restraints like this can be employed. Of course, cremation could theoretically be used, but this creates its own often worse problems.
What makes a person turn into an undead creature without intervention?
Three basic problems can occur. First, if a soul dies in a particular unpleasant way, it can be so disoriented in the afterlife that it is unable to find its way to a permanent abode. These lost souls wander the border ethereal trying to relive their life, too distressed and confused to move on. Gentle counseling can sometimes heal these lost souls, as can the intervention of a psychopomp.
Secondly, the souls of particularly evil persons can often linger in the border ethereal for malicious reasons, and whether disembodied or reinhabiting their decaying form seek to perform some evil that they were denied in life. Quite often, evil spirits will help arrange for this to happen, by thwarting psychopomps from completing their duty and there by spreading ruin and chaos. More rarely, a spirit can be lured by evil spirits to remain out of a sense of misplaced justice, which normally results in psychotic episodes and the tainting of that soul in the long run.
Thirdly, and most insidiously, necromantic energy lingers and accumulates in places where it is used. Over time, evil deeds and in particular evil spells (such as 'Animate Dead') leave behind necromantic pollution. When that pollution builds up, bodies will spontaneous animate as undead creatures with no intervention involved. Pretty much anywhere someone casts 'animate dead', there is a strong chance that any other bodies buried there will eventually reanimate as uncontrolled murderous undead. If you do it repeatedly, it's dead certain. This is one of the several reasons that most necromantic magic is inherently evil and banned from most right thinking communities.
On the other hand, some communities try to harness this power, going the other direction with it to deliberately create what amounts to necromantic toxic waste dumps.
Incidentally, this is the big problem with cementation undertaken outside of the bounds of sacred funeral rights. Mass burning leads to creating clouds of necromantic waste that taints everything it touches. All that ash ends up creating incorporeal undead that is far far worse than a few zombies or ghouls, and much harder for an average community to deal with. Catacombs are much safer in the long run, even if it does mean somewhere near the town you're likely to have a reservoir of necromantic taint and an undead problem.
Undertaker is a highly skilled profession in this setting. You don't want some random dummy responsible for interring the dead. The Guild of Undertakers is usually either entirely made up of oath sworn laity of some deity responsible for the dead, or else works in very close coordination with the cult thereof.
Likewise, the problem with grave robbery isn't just the theft, but that desecrating hallowed ground can have really nasty side effects.
