Consider the random walk on the integer number line, $\mathbb{Z}$, which starts at 0 and at each step moves $+1$ or $−1$ with equal probability. The probability for the event that "the first return to the origin occurs at epoch $2v$" is denoted by $f_{2v}$. By definition $f_0 = 0$.
In the end of section 3.3 (titled "The Main Lemma") of the book "An Introduction to Probability Theory and Its Applications" by W.Feller, it is proved that $f_2 + f_4 + \cdots = 1$ and remarked that
I am quite confused about this remark. Specifically, what does the "ultimate equalization of the fortune" mean? Furthermore, how is the probability (0.08) that no equalization occurs in 100 tosses obtained?
It means that there exists a (random) time $T$ which is (almost surely) finite and when the fortunes will be equal again.
Numerically, summing the probabilities that the first "equalization" occurs at time $2$ or at time $4$ or... or at time $100$ and taking the complement to $1$.