Absorbing Markov Chain Probabilities Can someone explain to me why the answer to the following question is not 1/2? (The given answer is 1/4, but, after trying various methods, including both argument by symmetry (each absorbing state must get 1/2, as the probability of being absorbed is 1, and the process is symmetric with respect to either state) and a more fulsome listing of the transition probabilities, I still get 1/2).
It's possible I don't understand the meaning of "probability that the process reaches the state AA and AA", but on the meaning "eventually" it should be 1/2, and on the meaning, in the next transition, it would be 1/16. So I am unable to find a meaning that gives us 1/4.
"Biologists can model a population's genotype distribution across generations with a Markov chain. Suppose there are two individuals with genotype Aa, and in each successive generation, two individuals are selected from the (numerous) offspring of the previous generation. These pairs of individuals form the possible states: AA and AA; AA and Aa; Aa and Aa; AA and aa; Aa and aa; aa and aa. Note that the first and last states listed are absorbing states. If the process reaches the state Aa and Aa, determine the probability that the process reaches the state AA and AA.
Recall that a child receives one allele (letter) from each parent, with a 50% probability of either allele being selected; for instance, the parents with genotype Aa will have a child with genotype AA with probability 25%, Aa with probability 50% and aa with probability 25%."

I, too, read the question as “What is the probability of reaching state $(\text{AA},\text{AA})$ given that the process starts in state $(\text{Aa},\text{Aa})$?” Your symmetry argument works and is borne out by an explicit calculation. There are some ways to get a probability of $\frac14$, by requiring that the process pass through state $(\text{Aa},\text{Aa})$ on its way to $(\text{AA},\text{AA})$, but that seems a bit of a stretch to me.
