Salvation does not come through mental faith alone. You have believed that Jesus Christ died for your salvation on Calvary and this faith places salvation within your reach.
Faith makes salvation possible.
Obedience to the Word of God brings salvation. This was not the end of the story. And, since repentance is to precede water baptism, there is no doubt that they also repented. Then in verse 34, the jailer received an experience that caused him to rejoice. The jailer's faith led him to meet the conditions of the gospel.