The great account of the Philippian jailer being saved shows us the Lord's work in this man's needy heart. Paul doesn't need to twist this man's arm instead he takes him directly to the answer and that is to believe in Jesus, John 1:12. We see that the church continues to grow despite suffering and persecution. The jailer is saved and in God's grace so is the rest of his family, which is reminiscent of Lydia's household. This is not a doctrine of all the family is saved because of one, as both scripture and experience tells us a different story. This is about the church of Philippi that the Lord has established.