No, that’s not how it happens at all.
We have 46 family trees. Half the 23 chromosome pairs come from each parent. There’s a tiny bit of mutation, there’s a tiny bit of gene-jumping, but these are of very low significance to the overwhelming stability of the inheritance process.
Each pair of grandparents do not contribute the same amount — the closest split would 12/11 or 11/12. Seven generations back, there are 64 great-great-great-great-grandparents, but only 46 chromosome halves to fell through to the next generation in each generation, so 18 of the g-g-g-g-grands don’t contribute anything to the current generation.
That doesn’t mean they’re not ancestors.
This is a great strength of sexual reproduction.
On paper, I know I’ve got common ancestors on both sides of my family. Thanks to DNA, I know I’m descended from my parents, and also know I didn’t get any matchy-matchy contributions from the common ancestors, because the parental DNA (that was passed to me anyway) isn’t identical on my chromosomes.
The fact that our Neandertal ancestors contribute as much as 3-4 percent (usually about 2-3 percent I think) means their chromosomes were a major source for all modern people, or it wouldn’t have persisted at the same percentage as each of our 4th-g-grands.
There are people around who think that chromosomes have ESP and play go fish with the genes to make sure those not yet conceived are exactly one quarter of each grandparent. Those people are nuts.
Look up genetic recombination or crossover.
The chromosome 3 that you give to your daughter is not likely to be the exact one you inherited from your father or your mother, but a mix of both.
https://en.wikipedia.org/wiki/Genetic_linkage