There are three passages of scripture that some have taken to mean that if one person in the household is saved, then all in the household are saved. Let’s take a look to see if that’s true:
Acts 11: 13, 14— And he reported to us how he had seen the angel standing in his house, and saying, “Send to Joppa and have Simon, who is also called Peter, brought here; and he will speak words to you by which you will be saved, you and all your household.”
This salvation promise was made by an angel to a Roman Centurion named Cornelius. The promise was made because God, knowing the future, saw that the household would be converted.
More importantly for us to understand, though, is that the household conversion only took place after all had had heard from Peter (Acts 10: 33). Peter preached to them the gospel. Cornelius and each household member were saved because, as is consistent with all salvation scripture, they individually heard and believed (faith comes by hearing the word of God).
Acts 16: 30, 31— And after he brought them out, he said, “Sirs, what must I do to be saved?” They said, “Believe in the Lord Jesus, and you will be saved, you and your household.”
This promise to a jailer, by the Apostle Paul and his companion, Silas, is a promise that is not limited. The jailer and his household, like everyone else throughout history, are saved upon belief in Jesus.
1 Corinthians 7: 14— For the unbelieving husband is sanctified through his wife, and the unbelieving wife is sanctified through her believing husband; for otherwise your children are unclean, but now they are holy.
The thing to note here is that the verse isn’t even talking about salvation. The topic is sanctification. Sanctification has to do with being set-aside as holy. In this case, that would be the institution of marriage.
I believe that this passage is connected to the Apostle Paul’s other teaching that a believer shouldn’t be unequally yoked with an unbeliever (2 Corinthians 6: 14). Paul recognizes, in 1 Corinthians 7: 14, that there are believers who are already married to unbelievers. For them, a distinction is made. He wants them and us to understand that as long as the unbelieving spouse consents to the marriage, the marriage remains holy based on the believer’s faith. The children, as a product of the marriage, are therefore in a sanctified relationship along with the parents.
That doesn’t mean the unbelieving spouse and children are saved, though. They are simply in a set-aside environment that gives them a chance to be saved by hearing the Word of God from the believer.
Viewed in the proper light, we see that the promises in the above passages are in line with every other verse concerning salvation. We each are accountable for our own actions. The faith of one cannot be applied to others who don’t believe.