How is property divided when a mother dies without a will?

Because your sister died without a will (the legal term is "intestate"), Missouri's "intestate succession" laws will determine who inherits her property.

Each state's law is different, but the closest relatives always inherit. So if your sister owned the house by herself, her children inherit it, in equal shares. (If she were survived by a spouse, he would also be entitled to a share of her property.) If she owned it with someone else, however, and they held title in joint tenancy, the surviving owner owns the entire property.