[CCWoody]

The short answer is that people accept by exercising the "free choice" ability which was given to them by God.

Pretend I am lost: If I say in the last moments of my life: "I accept Jesus into my heart, am I saved?"

[ShadowAce]

If you truly mean it, yes you are. If you make the confession just on the off-chance that Christianity may have a point after all, then no--you are not.

Was the thief on the cross saved at the last minute? Of course he was--Jesus Himself said so.

[FreedominJesusChrist]

God has given us a free will and we are not robots. If you truly are sorry for your sins, ask for forgiveness, and believe that Jesus died to take away your sins, then you are truly saved.

John 3:16-17 = For God so loved the world that he gave his one and only Son, that whoever believes in Him shall not perish but have eternal life. For God did not send his Son into the world to condemn the world, but to save the world through Him.

Notice that the Bible does not just say the few elect or those that have been specially chosen ahead of time?

[winstonchurchill]

If I say in the last moments of my life: "I accept Jesus into my heart, am I saved?"

Yes, oh, yes! As the much mis-used hymn has it, "Free at last, free at last, thank God Almighty I'm free at last."

Oh, how do we know? We have the word of Jesus Himself on the subject. To the thief on the cross: "I tell you the truth, today you will be with me in paradise." [Luke 23:43]