Not a biologist, but… to clarify: Genetics 101
You have 23 specific distinct PAIRS of chromosomes, which contain (almost) all the DNA that distinguishes you. One “pair” is XY if you are male, XX if female.
You got half of each pair from mom and half from dad.
These are specific and diferent - pair 1 is different from pair 2 is different from pair 3 etc. So you don’t get (for example) 3 of mom’s pair 3 and one of dad’s pair 4.
When making eggs or sperm for Junior, your gonads create those egg or sperm cells with half a load - one of the two from each pair of chromosomes.
So to pick a pair at random - say the 3rd pair - if they are both “West Europe” in a parent then Junior will definitely get a “West Europe” as part of their 3rd pair, plus whatever the mate contributes from their 3rd pair.
If the parent has a WE and a Brit as pair 3, then Junior will get either the WE or Brit. Pick only one. (Plus whatever the other parent contributes…)
So simplistically, if one chromosome controls red hair, Junior has a 50-50 chance of having red hair. If mama has a pair where both have red-hair chromosomes, Junior will definitely get a red-hair one.
Chromosomes duplicate when cells split, including when they produce eggs and sperm; during the process, the DNA spiral “unzips” like a zipper and collects matching molecules on the two pieces to create two DNA spirals. During this “copy” mistakes can happen - wrong piece, or the pieces “break” and fix and so part of one of the pair may attach to the other of the pair, or duplicate chunks may occur, or any of a dozen possible problems that create differences (transcription errors) in DNA can occur. 99% of the time this change is irrelevant, or dangerous to the point of reducing survivability, but sometimes it creates a better specimen of the species.
Two points -
This mixing (transcription error) of the two chromosomes can mean that the chromosome you pass on may not be 100% from one parent, but could actually be a mix.
The harmless or useful changes can accumulate over time creating differences that spread among the population over time - which is how different populations have somewhat different genes.
The rate these chromosomes change is also roughly regular, giving an approximate clock on how long it’s been since two people had a common ancestor.
Plus as mentioned, mitochondrial DNA - little DNA rings floating around in the cell but not in the nucleus - are only inherited from the mother via the egg, as sperm cells don’t have them. However, when the cell splits there’s no guarantee you have a copy of every different one left in the gel inside that cell - you may only get some…
So you can tell who is Thomas Jefferson’s male line descendant (or from his brother or uncle) based on the similar Y chromosome; you can tell who has the same all-female lineage based on mitochondria.
So the math for the first generation is simple - you got half your chromosomes from each parent. Beyond there, it gets fuzzy. of your dad’s 23 chromosomes, some came from his dad, some from his mom. Like flipping a coin 23 times, odds are close to 50-50; but in specific instances, anything can happen (plus that possibility of a transcription error meaning not completely). The only certainty is that with 23, it won’t be exactly 50-50. Go back enough generations and there’s a good chance you have nothing in common with perhaps one of your ancestors.
So the odds that all the genes you pass on to your offspring are from only one of your parents - well, the odds of flipping a coin 23 times and getting all heads is 2^23 or about 1 in 8.3 million.