It is theoretically possible, but the chances are so small as to be meaningless in the practical sense. If you are looking at just one marker or chromosomal region, you will see that two siblings will share no DNA about 1/4 of the time. That's just as you worked out when you wrote down the possible combinations (except you wrote AB instead of AC).
But that would have to happen over and over for siblings to show no match. It's hard to say how many "pieces" of DNA we should be using in our calculations because of recombination, so let's just take a simpler case of the 22 autosomes that always come in matching pairs. Imagine tossing a coin 22 times and coming up with all heads for one sibling and all tails for the other sibling. The formula for calculating the chance of that happening is 0.25 ^ 22 (raised to the 22 power). That's one in 17,592,186,044,416 (17.6 trillion).
Although I used the example of all heads and all tails, the same formula applies if you write down the sequence of 22 coin tosses for one sibling (HTTHTHHHT....). The chance of the sibling exactly duplicating that sequence is the same as all heads for one person and all tails for the other person.
Ann Turner
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