It is not all that unlikely that a parent will remarry after the death of a spouse, especially if they enjoyed a successful first marriage. Is it tacky to ask your parent what will happen to their assets if they don't survive their new partner? The unintentional disinheritance of adult children it not that uncommon – it happens because of the way most married couples own their assets - as, well, a couple. Regardless of what the will indicates, at death, if your parent died owning assets in joint accounts then those assets, by operation of law, completely bypass the will and go directly to the joint owner. Wills control only probate assets.
So, what's a probate asset? Probate assets are those that a person owns alone without a named beneficiary. The probate assets will pass in accordance with the terms of a person's will. However, if there's no will, the assets pass according to the state's laws of intestacy. Non-probate assets, on the other hand, include assets that a person owns jointly with another person, like a home or a joint checking accounts, and assets which designate a beneficiary. These are things like life insurance and retirement assets. As I mentioned, these aren't governed by the person's will or the laws of intestacy (unless the named beneficiary is the estate).
When a parent dies with probate assets, his estate would be required to be administered through the probate process in the county of his residence at the time of his death. If the father had a will, it's presented for probate. The person named in the will as the executor can qualify and obtain Letters Testamentary—these authorize him or her to act on behalf of the estate. If there isn't a will, an administrator is appointed. Generally, the spouse and then the children will have the first right to such an appointment. In either situation, everyone named in the will and all heirs at law (which include children) must be given notice of probate. They can request and receive a copy of the will. A person can choose to leave nothing to a spouse, but a spouse is typically entitled to claim an elective share. In many states, the spouse receives one third of a decedent's estate.
If the father died without a will and if there was no premarital agreement, then assets pass in intestacy where the surviving spouse will receive the first 25% under some state laws. The rest will pass to the children (and the children of deceased siblings), if any.
Reference: New Jersey 101.5 (March 29, 2016) "What to know about a second marriage and an inheritance"