Romans 9:11 KJVS
(For the children being not yet born, neither having done any good or evil, that the purpose of God according to election might stand, not of works, but of him that calleth;)

The terms promise, election, inheritance, and predestination are all talking about the same thing. The ninth chapter of the book of Romans is largely all about the election of God.

Romans 9:8 KJVS
That is, They which are the children of the flesh, these are not the children of God: but the children of the promise are counted for the seed.

The seed, of course, is Jesus Christ, the promised seed. And those that are in him are heirs according to the promise. The children of promise.

Galatians 3:14,16,18,29 KJVS
That the blessing of Abraham might come on the Gentiles through Jesus Christ; that we might receive the promise of the Spirit through faith. [16] Now to Abraham and his seed were the promises made. He saith not, And to seeds, as of many; but as of one, And to thy seed, which is Christ. [18] For if the inheritance be of the law, it is no more of promise: but God gave it to Abraham by promise. [29] And if ye be Christ’s, then are ye Abraham’s seed, and heirs according to the promise.

Inheritance, or being a heir it’s only through the promise seed. And those that received that inheritance were predestined to receive it, by the luck of the draw.

Ephesians 1:4-5,11 KJVS
According as he hath chosen us in him before the foundation of the world, that we should be holy and without blame before him in love: [5] Having predestinated us unto the adoption of children by Jesus Christ to himself, according to the good pleasure of his will, [11] In whom also we have obtained an inheritance, being predestinated according to the purpose of him who worketh all things after the counsel of his own will:

Definition for inheritance in verse 11.

Thayer’s Definition
1. to cast lots, determine by lot 1.
2. to choose by lot 2.
3. to allot, assign by lot

1 Corinthians 1:27-29 KJVS
But God hath chosen the foolish things of the world to confound the wise; and God hath chosen the weak things of the world to confound the things which are mighty; [28] And base things of the world, and things which are despised, hath God chosen, yea, and things which are not, to bring to nought things that are: [29] That no flesh should glory in his presence.