Because he asked Paul for instruction, a normal thing to do if you want to know more about something you believe is true.
Paul and Silas are imprisoned because they cast a demon out of a young woman who had said, “These men are the servants of the Most High God,
who proclaim to us the way of salvation.” The magistrates arrest them, beat them bloody and threw them naked into an inner prison cell put them in shackles and command the jailer to keep watch over them.
Paul and Silas proceed to sing hymns to God. Suddenly an earthquake shakes the prison, the shackles fall off all the prisoners. The jailer, fearing he will be executed because the prisoners we was supposed to be securing have surely escaped, draws his sword and is about to kill himself. Paul yells out, "Do yourself no harm, we're all here!" Whereupon the jailer asks Paul and Silas, "What must I do to be saved?"
31 So they said, “Believe on the Lord Jesus Christ, and you will be saved, you and your household.”
32 Then they spoke the word of the Lord to him and to all who were in his house.
33 And he took them the same hour of the night and washed their stripes. And immediately he and all his family were baptized. 34 Now when he had brought them into his house, he set food before them; and he rejoiced,
having believed in God with all his household.
Here we see that the jailer and his household were not believers until after they had heard the gospel preached to them.
So why do you think the jailer was already a believer when the scripture clearly says he was not?