another way to see it:
take different colored blocks (say 10 each color and label them with the #'s 1-10)
white will be Brad and black will be Brads wife
for each of their children, randomly pull out 5 blocks from each parents pile and write down the results (remember to put the blocks back before drawing for the next child). Each of there children will now be represented by 5 black and 5 white blocks (which may or may not be different #'s).
These children get married to unrelated people. Give each of the new spouses their own set/color of 10 blocks.
for their children (i.e Brads grandchildren) take 5 random blocks from their parent (who is brads child and who is now represented by 5 white and 5 black blocks) and 5 from the spouses set. This will represent their inherited DNA.
lets say you used Pink blocks for one of the spouses
theoretically the grandchild could be represented by:
5 white and 5 pink (and didn't inherit any of Brads DNA)
4 white 1 black and 5 pink
3 white 2 black and 5 pink
2 white 3 black and 5 pink
1 white 4 black and 5 pink
5 black and 5 pink (and didn't inherit and of Brad's wife's DNA)
now for the great grandchildren:
and example of one born that had both parents that were grandchildren of Brad:
lets say grandchild A marries grandchild B
A - has 3 white, 2 black and 5 pink as his DNA
B - has 1 white, 4 black and 5 green as her DNA
randomly draw 5 blocks from each of them and they could theoretically have a great grandchild with 5 pink and 5 green blocks and share no DNA with Brad
now take the grandchild who marries the Irish girl:
he could have 5 blocks white and 5 blocks Blue
his irish wife has her 10 blocks (lets say 5 yellow and 5 purple that she got from her parents)
their child (Brads great grandchild) could end up with 5 white and 5 yellow and have more of Brads DNA then the other great grandchildren.
That's the randomness of DNA that makes it so hard to pinpoint the exact level of "cousinship"
and its why 2 siblings can take DNA tests and have different people on their lists of matches.