|MadSci Network: Genetics|
Thanks for your question. While it is theoretical, that does not make it unimportant.
There is indeed a "very very small chance" that a cross between a poodle and a poodle x chihuahua hybrid would produce "a dog that is genetically a poodle". All the would need to happen would be for the poodle x chihuahua to produce gametes (doesn't matter whether they are egg or sperm) that receive only the poodle contributions from the parent. And there is obviously a very very small chance that this could occur.
More interesting, I think, is the whole question of what constitutes a genetic poodle, and how one might distinguish a genetic poodle from a dog that looks exactly like a poodle. In other words, what is the relationship between the phenotype and the genotype for dog breeds?
There are abstracts to a couple of interesting scientific papers on the subject here and here.
Moderator's note: Ignoring recombination of chromosomes during meiosis (which makes the math more complicated), you can get a rough estimate of the frequency of all poodle gametes produced by the poodle x chihuahua hybrid as follows. The probability of the poodle x chihuahua producing an all poodle gametes is the probability that when each chromosome pair is segregated during meiosis, a given gamete receives the poodle parental chromosome each time. Since chromosomes segregate randomly and independantly (mendel's laws), this is simply (1/2) to the power of n, where n is the number of chromosomes. Dogs have 39 chromosome pairs (humans have 47 pairs), thus the probability of getting a pure poodle gamete is (1/2)^39 = one in 2^39 or one in 5.5 x 10^11, or pretty small. Considering that dogs generally have much much less than 10^11 offspring, this is pretty unlikely to happen.
For more about meiosis (how these gametes are formed and visuals on the chromosomes segregating), see here
Try the links in the MadSci Library for more information on Genetics.