Sorry that is not found in the passage. You adding your theology to itWhen Paul exhorted the jailer to believe in the Lord Jesus Christ that he might be saved, God enabled the jailer to believe Life and power is conveyed to the soul in Gospel Commands and Exhortations.
Believing in Christ is given/caused by Grace Phil 1:29
29 For unto you it is given in the behalf of Christ, not only to believe on him, but also to suffer for his sake;
The word given charizomai which comes from the word grace and means:
to give graciously, give freely, bestow so Grace is bestowed to cause believing. Like when sight is given Lk 7 21
21 And in that same hour he cured many of their infirmities and plagues, and of evil spirits; and unto many that were blind he gave sight.
More: The word given charizomai in Phil 1:29 is the exact same word freely give in Rom 8:32
He that spared not his own Son, but delivered him up for us all, how shall he not with him also freely give us all things?
So all for whom Christ was delivered up for, with Him they will be given freely with Him to believe on Him.
The phillipians were granted faith in this manner
Acts 16:9–12 (KJV 1900) — 9 And a vision appeared to Paul in the night; There stood a man of Macedonia, and prayed him, saying, Come over into Macedonia, and help us. 10 And after he had seen the vision, immediately we endeavoured to go into Macedonia, assuredly gathering that the Lord had called us for to preach the gospel unto them. 11 Therefore loosing from Troas, we came with a straight course to Samothracia, and the next day to Neapolis; 12 And from thence to Philippi, which is the chief city of that part of Macedonia, and a colony: and we were in that city abiding certain days.
Paul received a vision and as a result went and preached the gospel
as scripture states
Romans 10:14–17 (KJV 1900) — 14 How then shall they call on him in whom they have not believed? and how shall they believe in him of whom they have not heard? and how shall they hear without a preacher? 15 And how shall they preach, except they be sent? as it is written, How beautiful are the feet of them that preach the gospel of peace, and bring glad tidings of good things! 16 But they have not all obeyed the gospel. For Esaias saith, Lord, who hath believed our report? 17 So then faith cometh by hearing, and hearing by the word of God.