The text of the story calls this answer into question. Paul's response is:
Acts 16: 29 The jailer called for lights, rushed in and fell trembling before Paul and Silas. 30 He then brought them out and asked, "Sirs, what must I do to be saved?" 31 They replied, "Believe in the Lord Jesus, and you will be saved--you and your household."
The scripture uses the future tense to describe the sequence between believing and salvation. In short, it appears to be saying, "Believe and AFTERWARDS you WILL BE saved."
I'm not asking what you believe at this point. I'm simply asking if you can see how, based on this text, a general reader could come to the conclusion that believing precedes salvation?
Sure. Since the Bible was written for people it's only natural that much of it is from man's POV.