What is the probability that a descendent carries even at least one gene from the "Most Recent Common Ancestor"?
The human genome has only 20,000 protein-coding genes (which comprise 1-2% of human DNA). I'll be generous and use the 1% figure to assume that human DNA has about 2e6 "genes" (protein-coding or otherwise).
After 100 generations:
Probability any one gene comes from MRCA: 1/(2^100).
Probability any one gene does not come from MRCA = 1 - 1/(2^100)
Probability NO gemes come from MRCA = (1 - 1/(2^100))^(2^6) = 0.9999999999999999999999999999495129020658552444536494
Probability at least one gene comes from MRCA = 5.05 × 10^-29
I think that you have confused mutation with gene in your initial assertion.
There is probably only a few percent of variation between all of your 2^100 ancestors genomes (the majority of whom will be related), so protein coding regions of your genome and your MRCA are actually likely to be very similar.
Now if that MRCA had a specific mutation, the chances of your inheriting that specific mutation are probably low, but depend on that mutations prevalence in the population as well as whether the mutation is deleterious or not.
100 generations is very few in terms of changes within the genetic make up of a population under normal circumstances.
>What is the probability that a descendent carries even at least one gene from the "Most Recent Common Ancestor"?
If you think about it, its very nearly 100%. The fact that he is the MRCA means that a particular gene of his conferred an enhanced survival advantage. This enhanced gene is likely to exist in all of humanity currently living.
The human genome has only 20,000 protein-coding genes (which comprise 1-2% of human DNA). I'll be generous and use the 1% figure to assume that human DNA has about 2e6 "genes" (protein-coding or otherwise).
After 100 generations:
Probability any one gene comes from MRCA: 1/(2^100). Probability any one gene does not come from MRCA = 1 - 1/(2^100) Probability NO gemes come from MRCA = (1 - 1/(2^100))^(2^6) = 0.9999999999999999999999999999495129020658552444536494
Probability at least one gene comes from MRCA = 5.05 × 10^-29