Legal Question in Wills and Trusts in Illinois

Assuming he didn't leave a Will stating otherwise, assets in his sole name would pass:

1/3 to Daughter 2

1/3 to Daughter 3

1/3 among Daughter #1's surviving children (1/9 to each of the 3)

This is the Illinois "per stirpes" treatment.

Just to add to the above answer - any assets in which in names a joint tenant with right of survivorship or in which he specified a beneficiary are not part of his Estate and do not figure into this calculation. Also with respect to the 1/3 which will go to the children of D#1 - in the event you find Mr. Gottlieb's answer a bit confusing as I do, he is basically saying that the 1/3 of the total Estate that goes to the surviving children is then divided into thirds with each child receiving 1/3 of the 1/3 which would have gone to D#1 had she survived, hence the reference to 1/9th of the total Estate.

I agree generally with the earlier answers, but your use of the term "beneficial interest" suggests to me that you are looking at something in particular that may include terms that would govern. As pointed out already, there are assets that would not be governed by the state intestacy laws, for example, assets held in trust for which a proper distribution is provided. I suggest you get more direct probate advice.