This is why Paul's reply to the jailer was a correction. The jailer was essentially asking Paul how create his salvation. The jailer said, "What must I do to be saved." The word do means "to create" or "to make."
It also infers simply doing something. Like believing, which was Paul's answer.
Paul disaffirmed the jailer's notion that he must do something by saying, "Believe on the Lord Jesus Christ and you shall be saved."
Which affirmed the jailer's question. What the jailer
MUST DO is to believe on the Lord Jesus Christ to be saved.
We see the exact same concept in a number of verses:
Mark 16:16
16 " He who has believed and has been baptized shall be saved; but he who has disbelieved shall be condemned.
Luke 8:12
12 "Those beside the road are those who have heard; then the devil comes and takes away the word from their heart, so that they will not believe and be saved.
Acts 4:12
12 "And there is salvation in no one else; for there is no other name under heaven that has been given among men by which we must be saved."
Acts 11:14
14 and he will speak words to you by which you will be saved, you and all your household.'
Acts 16:31
They said, Believe in the Lord Jesus, and you will be saved, you and your household."
Rom 10:9, 10
9 that if you confess with your mouth Jesus as Lord, and believe in your heart that God raised Him from the dead, you will be saved; 10 for with the heart a person believes, resulting in righteousness, and with the mouth he confesses, resulting in salvation.
Rom 1:16
16 For I am not ashamed of the gospel, for it is the power of God for salvation to everyone who believes, to the Jew first and also to the Greek.
Eph 2:8 For by grace you have been saved through faith; and that not of yourselves, it is the gift of God;
2 Tim 3:15
15 and that from childhood you have known the sacred writings which are able to give you the wisdom that leads to salvation through faith which is in Christ Jesus.
1 Peter 1:5
5 who are protected by the power of God through faith for a salvation ready to be revealed in the last time.
1 Peter 1:9 obtaining as the outcome of your faith the salvation of your souls.
2 Thess 2:13 But we should always give thanks to God for you, brethren beloved by the Lord, because God has chosen you from the beginning for salvation through sanctification by the Spirit and faith in the truth.
All of these verses clearly link believing with being saved or obtaining eternal life.