And by His will, we do.
You seem incredulous that anyone would presume their children are among God's family. Scripture tells parents to believe exactly that...
"He will bless them that fear the LORD, both small and great. The LORD shall increase you more and more, you and your children." -- Psalm 115:13-14
"For the promise is unto you, and to your children, and to all that are afar off, even as many as the LORD our God shall call." -- Acts 2:39
"And again, I will put my trust in him. And again, Behold I and the children which God hath given me." -- Hebrews 2:13"She is not afraid of the snow for her household: for all her household are clothed with scarlet." -- Proverbs 31:21
I wasn't aware that was part of TULIP, or at least your views. Sorry.
Is the reverse true for children born to the non-elect?