He tells the jailer, believe in Christ and you will be saved. You and your oikos. (He could have omitted 'and your oikos') and just go on with the narrative. Up until recently these verses would have universally been understood to include children (and servants). No one in the Levant or Greece or Egypt especially 1800 years ago would ever suppose Paul only meant a household full of adults. The Oikos was managed by the master of the house and internally managed by the wife cooperatively, and all other fell in line. The modern day interpretation of Mark 16:31 as consisting of only adult children would never pass muster in Paul's day. Paul's whole intent is to show the oikos is included. this is no different than Abraham who was required to circimcise ALL males of his household regardless:
Then God said to Abraham, “As for you, you must keep my covenant, you and your descendants after you for the generations to come. 10 This is my covenant with you and your descendants after you, the covenant you are to keep: Every male among you shall be circumcised. 11 You are to undergo circumcision, and it will be the sign of the covenant between me and you. 12 For the generations to come every male among you who is eight days old must be circumcised, including those born in your household or bought with money from a foreigner—those who are not your offspring. 13 Whether born in your household or bought with your money, they must be circumcised. My covenant in your flesh is to be an everlasting covenant. 14 Any uncircumcised male, who has not been circumcised in the flesh, will be cut off from his people; he has broken my covenant.” (Genesis 17)
Now one of the biggest fallacies of those who claim sola scripture is the non-bibilical belief that children cannot believe in Christ. Where in the bible does it teach this? On the contrary scripture teaches children are naturally inclined believers in Christ:
And whoso shall receive one such little child in my name receiveth me.6 But whoso shall offend one of these little ones which believe in me, it were better for him that a millstone were hanged about his neck, and that he were drowned in the depth of the sea. (Matt 18:5-6)
15 And when the chief priests and scribes saw the wonderful things that he did, and the children crying in the temple, and saying, Hosanna to the son of David; they were sore displeased,
16 And said unto him, Hearest thou what these say? And Jesus saith unto them, Yea; have ye never read, Out of the mouth of babes and sucklings thou hast perfected praise? (Matt 21)