Why is the probability of fixation of an allele equal to its frequency?
Several proofs are given here (p. 9). My favorite comes from the genealogical argument:

Consider the situation where there are $2N$ alleles: $A_1$, $A_2$, $A_3$ ... $A_{2N}$.

By the genealogical argument, we may state that at $t = \infty$, all alleles at this locus will be direct descendants of one particular allele present at $t = 0$.

Allelic variants at this locus are selectively neutral, so $Pr(A_{1fix})$ $=$ $Pr(A_{2fix})$ $=$ $Pr(A_{3fix})$ $= ... =$ $Pr(A_{2Nfix})$. For any given allele present at $t = 0$, the probability of fixation is therefore $\frac{1}{2N}$.

Now define allelic variants $A$ and a as complementary, non-overlapping groups of the initial alleles, such that $n_A$ + $n_a$ $=$ $2N$. From the above, the probability of fixation of an allele within group $A$ is $n_A * \frac{1}{2N} = \frac{n_A}{2N} = p_0$.