Is there any connection between God’s grace and man making a free-will choice to believe?
GRACE means God's undeserved favor towards man.
“But we believe that through the grace of the Lord Jesus Christ we shall be saved” (Acts 15:11)
“… those who had believed through grace” (Acts 18:27)
“… His grace, which is able to … give you an inheritance among all those who are sanctified” (Acts 20:32)
“… being justified freely by His grace” (Romans 3:24)
“… those who receive abundance of grace and of the gift of righteousness will reign in life” (Romans 5:17)
“… there is a remnant according to the election of grace” (Romans 11:5)
“In Him we have redemption through His blood, the forgiveness of sins,
according to the riches of His grace” (Ephesians 1:7)
Ephesians 2:
1 And you He made alive, who were dead in trespasses and sins,
2 in which you once walked … according to the prince of the power of the air …
4 But God, who is rich in mercy, because of His great love with which He loved us,
5 even when we were dead in trespasses, made us alive together with Christ
(by grace you have been saved),
6 and raised us up together, and made us sit together in the heavenly places in Christ Jesus,
7 that in the ages to come He might show the exceeding riches of His grace
in His kindness toward us in Christ Jesus.
8 For by grace you have been saved through faith, and that not of yourselves; it is the gift of God,
9 not of works, lest anyone should boast.
10 For we are His workmanship, created in Christ Jesus for good works,
which God prepared beforehand that we should walk in them.
“… God, who has saved us and called us with a holy calling, not according to our works,
but according to His own purpose and grace, which was given to us in Christ Jesus
before time began” (2 Timothy 1:8-9)
“For the grace of God that brings salvation has appeared to all men” (Titus 2:11)
Could this mean that all men have seen what the grace of God can do for others?
Or, does this mean that all men (who are God's elect) have experienced God's saving grace?
“… not by works of righteousness which we have done, but according to His mercy He saved us,
through the washing of regeneration and renewing of the Holy Spirit, whom He poured out on us
abundantly through Jesus Christ our Savior, that having been justified by His grace
we should become heirs according to the hope of eternal life.” (Titus 3:5-7)
“Of this salvation the prophets have inquired and searched carefully,
who prophesied of the grace that would come to you” (1 Peter 1:10)
“… the God of all grace, who called us to His eternal glory by Christ Jesus” (1 Peter 5:10)
Yes, you are correct ...
there is absolutely no connection between God's grace and man's choice to believe.