No non-believer genuinely wants or is able to seek after God's righteousness (Jer. 17:9; Mark 7:21-23; Eph. 2:2; Eph. 2:4-5; Titus 3:5; John 3:19; Rom. 3:10-12; 5:6; 6:16-20; Eph. 2:1,3;1 Cor. 2:14), hence, thanks be to God for election (John 6:37; John 6:39; John 10:29; John 17:11-12; John 17:9;John 17:22; John 18:9). Once regenerated (born anew), the non-believer instantaneously will genuinely want, will be able, and will not not believe.
Election is unto an end, that is salvation, not to salvation. In other words, under ordinary circumstances, the elect will come to faith from the hearing of the Good News, being regenerated, now able and wanting to believe, per God's gift of faith, then irrevocably believing.
Faith must include "choice," if by that is meant an act of the will. Faith is simply the man believing. Man thinks, wills, and affects. Therefore faith is defined in terms of knowledge, assent, and trust. It is certain that man cannot will anything spiritually good in a fallen and unregenerate condition, but it is equally certain that when a man is regenerated he is enabled to believe in Our Lord for salvation. God does not believe for him. God works in him to will and to do of His good pleasure (Philippians 2:13).
Hebrews 11:1 provides a definition of what faith does rather than what faith is. It is describing the function rather than the nature of faith. It is regrettable that it has been translated by some in subjective terms when it is indicating something that is objective.
That Hebrews 11:1 speaks of the function of faith, not the nature of it, is clear from the context, which is believing to the saving of the soul, Hebrews 10:38-39.
AMR