... the jailer was not asking what should he do in order to indicate he was elect. He was asking what he must to in order to be saved.
You've assumed the jailer was
un-elect, yes ?
A little known fact about Acts 16 is that Paul and Silas were likely
psalm-singing Psalm 119... and possibly fasting that night...
Psalm 119,
"At midnight I will rise to give thanks unto thee because of thy righteous judgments. I am a companion of all them that fear thee, and of them that keep thy precepts. The earth, O Lord, is full of thy mercy: teach me thy statutes. ... They had almost consumed me upon earth; but I forsook not thy precepts.... Thy faithfulness is unto all generations: thou hast established the earth, and it abideth. ... Thou puttest away all the wicked of the earth like dross: therefore I love thy testimonies. ... Lord, I have hoped for thy salvation, and done thy commandments." (Ps. 110:62-63, 87, 90, 119, 166)
Acts 16,
"And at midnight Paul and Silas prayed, and sang praises unto God: and the prisoners heard them. And suddenly there was a great earthquake, so that the foundations of the prison were shaken: and immediately all the doors were opened, and every one's bands were loosed. ... And when it was day, the magistrates sent the serjeants, saying, Let those men go." (25-26, 35)
God heard them as he heard David. The earth literally trembled. And as the Psalmist had
hoped for salvation these men were saved.
It may also be related to God's true 'fast' in Isaiah, his true intent and will, to free the oppressed... and it seems he jailers were using unjust corporal punishment (maybe lashes),
"They have beaten us openly uncondemned" (Paul, v. 37) and God hates injustice and iniquity,
"hands that shed innocent blood" (Pro. 6:16-19)
Isaiah 58,
"Wherefore have we fasted, say they, and thou seest not? wherefore have we afflicted our soul, and thou takest no knowledge? ... Behold, ye fast for strife and debate,... Is not this the fast that I have chosen? to loose the bands of wickedness, to undo the heavy burdens, and to let the oppressed go free, and that ye break every yoke? Is it not to deal thy bread to the hungry, and that thou bring the poor that are cast out to thy house? Then shall thy light break forth as the morning, and thine health shall spring forth speedily: and thy righteousness shall go before thee; the glory of the Lord shall be thy reward. (Isa. 58:3-8)
So it's possible those men who were unshackled by God were already elect men. Similar to the Lord visiting Jericho and isolating Zacchaeus in the tree, ignoring the masses. He was there to save him:
"This day is salvation come to this house... For the Son of man is come to seek and to save that which was lost." (Lk. 19:9-10)
In the same way, God had Paul save many. Thus, it's likely the jailer and "everyone" else there were already elect by God. - There are no coincidences with God.
Another point to remember is God's promise to
save households, not only
individuals (as we focus on nowadays as a highly
individualistic society), including servants and slaves like Hagai, even the entire city of Nineveh. "And they said, Believe on the Lord Jesus Christ, and
thou shalt be saved, and thy house." (Acts 16:31)
This
collective salvation whether household / tribal / denominational / covenantal / national is something Presbyterians and other Calvinists emphasise greatly. Baptistic churches have little awareness of this. i.e. Baptists are typically very new converts, 1st or 2nd generation believers, while Presbyterians tend to be born into the faith being the 3rd, 4th, 5th, plus, generation. I know Huegenots having come from the French persecution, also Scottish Presbyterians dating to the 1850s Disruption and earlier to John Knox. Hence, these Calvinists have a much greater emphasis on the 'doctrines of grace' / 'election', since in their experience God has genuinely loved them and their ancestors as he has promised, collectively, generationally, denominationally, covenantally and nationally.
"For
the promise is unto you, and to your children, and to all that are afar off, even as many as the Lord our God shall call." (Acts 2:39) "And shewing mercy unto
thousands of them that love me, and keep my commandments." (Ex. 20:6)
Something to consider...
For
new converts like the Philippian jailer, he would be in a similar position to the
Thief on the Cross, so to "get into that state in which you are free from a hell-bound fate", he would need to do the same.
So the answer to your question would be to
"Repent and believe the Gospel" (Mk 1:15)
... and or
"Repent, and be baptized every one of you in the name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Ghost." (Acts 2:28)
Repentance, however, isn't applicable to someone who's
always believed, having been brought up in the faith, having been faithful for as long as they can remember. So, this part is often neglected by Presbyterians... Only an apostate can changed their mind / way to turn back to God... similar to the Jews who were repenting at the Jordan River on account of John's ministry. They had to be mourning something, to be in a state of regret, remorse about the 400-year silence, etc.
For Calvinists this experience would be an aspect of election... an aspect of God's 'grace'... "ALL those whom God hath predestinated unto life... come most freely, being
made willing by his grace.h" (WCF 10.1) -- Thanking and praising God regardless (not boasting of individual works or free-will).
However, in other circumstances, a Calvinist
does have an
introspective process (done individually, mostly by one's free-will), but like the above, credit is given to 'Spirit of Christ' and the Holy Spirit.
"
Their ability to do good works is not at all of themselves, but wholly from the Spirit of Christ.m And, that they may be enabled thereunto, besides the graces they have already received, there is required an actual influence of the same Holy Spirit to work in them to will and to do of his good pleasure:n yet are they not hereupon to grow negligent, as if they were not bound to perform any duty, unless upon a special motion of the Spirit; but they ought to
be diligent in stirring up the grace of God that is in them.o" (WCF 16.3)
This however isn't for 'salvation', more so for a sense of 'assurance of salvation'. To gauge one's maturity in faith, and staying on track towards salvation, eventually. - 'Always hoping' etc
This is where the cyclical reasoning comes to play...
"
THEY whom God hath accepted in his Beloved, effectually called and sanctified by his Spirit, can neither totally nor finally fall away from the state of grace; but shall certainly persevere therein to the end, and be eternally saved.a" (WCF 27.1)
Proof:
a Phil. 1:6. Being confident of this very thing, that
he which hath begun a good work in you, will perform it until the day of Jesus Christ. 2 Pet. 1:10. Wherefore the rather, brethren, give diligence to make your calling and election sure: for if ye do these things, ye shall never fall. John 10:28. And I give unto them eternal life, and they shall never perish, neither shall any pluck them out of my hand. Ver. 29.
My Father, which gave them me, is greater than all: and none is able to pluck them out of my Father’s hand. 1 John 3:9. Whosoever is born of God doth not commit sin; for his seed remaineth in him: and he cannot sin, because he is born of God. 1 Pet. 1:5. Who are kept by the power of God through faith unto salvation, ready to be revealed in the last time. Ver. 9. Receiving the end of your faith, even the salvation of your souls.
Thus, the un-elect
cannot profess to be Christian...
"Others not elected, although they may be called by the ministry of the word,p and may have some common operations of the Spirit,q yet they never truly come unto Christ, and therefore cannot be saved;r much less can men not
professing the Christian religion be saved in any other way whatsoever, be they ever so diligent to frame their lives according to the light of nature, and the law of that religion they do profess;s and to assert and maintain that they may, is very pernicious, and to be detested.t" (WCF 10.4)
The verb choice here ('to profess') is quite important in understanding Calvinism...
Unlike Baptists who 'declare' their new belief and religious affiliation, to a Presbyterian the word 'declare' is preposterous (since man's free-will is limited, he's not God). Instead, one can only
respond to God's grace / election, and when ready make a 'confession' (at age 16+, publicly) of their belief in Christ (exactly as the Jailer, I suppose !). Similar to the Lord's baptism, minus the baptism ritual. - New adult converts do this too, plus baptism, all at once.
In this way, at the
very minimum, a Presbyterian believes it's OK to have a
nominal belief, i.e. merely saying 'I'm a Christian' (in an informal setting), since the very ability to say that is
assumed to come from God. -- Nominalism is of course is a major problem in Christendom and 'Christian' cults.
a Phil. 1:6. Being confident of this very thing, that
he which hath begun a good work in you, will perform it until the day of Jesus Christ. 2 Pet. 1:10. Wherefore the rather, brethren, give diligence to make your calling and election sure: for if ye do these things, ye shall never fall.
In saying that, however, people born into the church are
not presumed to be Christian, elect or saved,
until they make a 'confession of faith' (whenever they choose to do so). They sort of are 'elect' or 'saved' in a general sense of God's 'grace' keeping them grounded, being in God's general grace ('raising the sun on the good and the wicked'), and various protections of government and church etc, however, their rebellion or misadventures are noted to be 'disruptive' to themselves (I don't think they can theoretically damn themselves according to the WCF though. Again, this is God's prerogative). It's a hopeful system that is waiting for people to repent, grow, and mature... It's not so dogmatic or damning to their own (apart from church discipline and monitoring 'family worship' etc, which the stricter dogmatic authoritarian churches do, like Calvin's did).
Also worth noting is that
Presbyterians welcome all Christians (and all people) to worship together, indiscriminately. That is, while Presbyterian / Calvinist doctrine is reserved only for elders (church governance and leadership), people in the public are
not discriminated against and not forced to say or do anything before joining in worship. This is radically different to Baptists or Quakers for instance who won't let someone through the doors if they look funny, maybe smoking, or smelling like alcohol.
Also depending on the church, while the congregation / members are
taught Calvinistic doctrines, they are
not expected or obliged to become Presbyterian / Calvinist in their belief... There is no legal requirement as far I'm aware... Hence, many congregations are mixed with Baptists, Methodists, Anglicans, Wesleyans, and the occasional Lutheran... E.g. You don't
have to get baptised, or confess, or say or do anything. You can just turn up and they will (should) look after you. - However,
some congregations will bar you from sacraments until they've assessed you, again being the more dogmatic and authoritarian ones. But a true church will be open for
all (whether for Communion, baptism, marriage, housing, or whatever), as the Gospel is for
all. An open invite.