...A family member recently died ... his probation was closed....
Yes. When one dies, he cannot change his state or standing with God. His probation does close.
Heb 9:27 And as it is appointed unto men once to die, but after this the judgment:
2 Cor 6:2 (For he saith, I have heard thee in a time accepted, and in the day of salvation have I succoured thee: behold, now is the accepted time; behold, now is the day of salvation.)
...where did the belief in probations being closed come from?...
The Bible. Jesus.
Matt 25:10 And while they went to buy, the bridegroom came; and they that were ready went in with him to the marriage: and
the door was shut.
Luke 13:25 When once the master of the house is risen up, and
hath shut to the door, and ye begin to stand without, and to knock at the door, saying, Lord, Lord, open unto us; and he shall answer and say unto you, I know you not whence ye are:
Rev 22:11 He that is unjust, let him
be unjust still: and he which is filthy, let him
be filthy still: and he that is righteous, let him
be righteous still: and he that is holy, let him
be holy still. 12 And, behold, I come quickly; and my reward is with me, to give every man according as his work shall be.
Daniel 12:1 And at that time shall Michael stand up...
Luke 17
26 And as it was in the days of Noe, so shall it be also in the days of the Son of man.
27 They did eat, they drank, they married wives, they were given in marriage, until the day that Noah entered into the ark, and the flood came, and destroyed them all.
28 Likewise also as it was in the days of Lot; they did eat, they drank, they bought, they sold, they planted, they builded;
29 But the same day that Lot went out of Sodom it rained fire and brimstone from heaven, and destroyed them all.
30 Even thus shall it be in the day when the Son of man is revealed.
31 In that day, he which shall be upon the housetop, and his stuff in the house, let him not come down to take it away: and he that is in the field, let him likewise not return back.
... As a child, I lived in total fear that at any moment my probation would be closed.
Your probation could close if you died or if Jesus was about to come. Perfect love casts out fear. People who are right with God welcome Jesus' coming. It is sinners who are sinning that are afraid.
Isa 25: 9 And it shall be said in that day, Lo, this is our God;
we have waited for him, and he will save us: this is the LORD; we have waited for him,
we will be glad and rejoice in his salvation.
Rev 6:15And the kings of the earth, and the great men, and the rich men, and the chief captains, and the mighty men, and every bondman, and every free man,
hid themselves in the dens and in the rocks of the mountains; 16 And said to the mountains and rocks, Fall on us, and
hide us from the face of him that sitteth on the throne, and from the wrath of the Lamb: 17 For the great day of his wrath is come; and
who shall be able to stand?