Then he (the jailor) called for a light, and sprang in, and came trembling, and fell down before Paul and Silas,
And brought them out, and said, Sirs, what must I do to be saved?
The jailor asked Paul and Silas what he must do for his own salvation.
He didn’t ask about anyone else’s salvation.
But Paul's and Silas’ response was:
And they said, Believe on the Lord Jesus Christ, and thou shalt be saved, and thy house.
The jailor’s family would be saved along with him – even though the jailor didn’t ask about them.
And the LORD said unto Noah, Come thou and all thy house into the ark; for thee have I seen righteous before me in this generation.
Again, the father of the family, Noah, was righteous before the Lord and thus had the legal right to bring all of his house to safety.
Fathers are expected to lead.
God knew Abraham would lead his family:
For I know him, that he will command his children and his household after him, and they shall keep the way of the LORD, to do justice and judgment;
Joshua made the decision for his family.
And if it seem evil unto you to serve the LORD, choose you this day whom ye will serve; whether the gods which your fathers served that were on the other side of the flood, or the gods of the Amorites, in whose land ye dwell: but as for me and my house, we will serve the LORD.
Caleb led his family to the promised land:
Save Caleb the son of Jephunneh; he shall see it, and to him will I give the land that he hath trodden upon, and to his children, because he hath wholly followed the LORD.
1 Timothy 3:5
(For if a man know not how to rule his own house, how shall he take care of the church of God?)
In contrast, fathers’ bad decisions can lead the entire family to hell.
And he spake unto the congregation, saying, Depart, I pray you, from the tents of these wicked men, and touch nothing of theirs, lest ye be consumed in all their sins.
So they gat up from the tabernacle of Korah, Dathan, and Abiram, on every side: and Dathan and Abiram came out, and stood in the door of their tents, and their wives, and their sons, and their little children.
…And the earth opened her mouth, and swallowed them up, and their houses, and all the men that appertained unto Korah, and all their goods.
They, and all that appertained to them, went down alive into the pit, and the earth closed upon them: and they perished from among the congregation.
…I the LORD thy God am a jealous God, visiting the iniquity of the fathers upon the children unto the third and fourth generation of them that hate me