A biological population comprises $N$ individuals that reproduce to generate N individuals in the next generation. (Due to the random sampling of the 2 alleles of the offsprings from 2 parents we can assume that the frequency of an allele should remain constant on average.) Every individual can carry $2$ different alleles (gene variants) out of a multitude of possible alleles. Thus, there are $2N$ alleles in the population.
Now we look at one particular allele $A$ and we know that the relative frequency of that allele $A$ is $p_A^{t_1}$ in generation $t_1$. Now we wait $k$ generations and measure in the generation $t_k$ the frequency of the allele $A$ again: $p_A^{t_k}$.
My question is: How can I calculate the probability for getting $p_A^{t_k}$ if the relative frequency of $A$ was $p_A^{t_1}$ $k$ generations before?
Let's further assume that every allele has the probability $\mu$ (mutation rate) to be transformed into a new allele in every generation.
Sub question: What is the probability for the change from $p_A^{t_1}$ to $p_A^{t_k}$ after $k$ generations now?