There has to be some kind of probate to pass title to property as well as to notify any creditors of the death.
There are a variety of kinds of probate depending on how much property is involved and whether there are any debts.
Avoiding probate is normally done by handling things before death, by giving them to the person who is going to receive them, making any accounts payable on death, etc.
Everything that the husband owned or owned a share in has to be taken into account. That means the house, any vehicles, any accounts, stocks, bonds, etc.
In addition, the probate has to be done if there are any creditors of the estate including any monies owed due to the "last illness".
Usually the best thing to do is make a list of the debts and of the assets and sit down with a local lawyer that does probate work and let them look over the lists and ask questions. They can ten tell you whether you need to do a probate and, if so, whether it can be an informal probate (such as an affidavit of heirship) or whether it has to be a full probate. If the children object at all then the affidavit of heirship isn't likely to be an option.
Also, if you own any real estate of any kind the only way to pass title and clear up the chain of title is through some kind of probate. Without that you can't sell or get the property refinanced and, depending on the language contained in the mortgage, the lender can even declare the mortgage in default.
Please ask any follow up questions in this thread.