Note: This situation deserves some additional considerations which put the problem in a different light.
The following is from chapter III: Fluctuations in Coin Tossing and Random Walks of the classic An Introduction to Probability Theory and Its Applications, Vol. I by W. Feller.
(W. Feller): For example, in various applications it is assumed that observations on an individual coin-tossing game during a long time interval will yield the same statistical characteristics as the observation of the results of a huge number of independent games at one given instant. This is not so.
He continues with
According to widespread beliefs a so-called law of averages should ensure that in a long coin-tossing game each player will be on the winning side for about half the time, and the lead will pass not infrequently from one player to the other.
But, in fact this is wrong and contrary to the usual belief the following holds:
With probability $\frac{1}{2}$ no equalization occurred in the second half of the game regardless of the length of the game. Furthermore, the probabilities near the end point are greatest.
The reasoning is based upon the Arcsine law for last visits (see e.g. Vol 1, ch.3, section 4, Theorem 1 in W. Feller's book): The probability that up to and including epoch $2n$ the last visit to the origin occurs at epoch $2k$ is given by
\begin{align*}
\alpha_{2k,2n}=\frac{1}{4^n}\binom{2k}{k}\binom{2n-2k}{n-k}
\end{align*}
Since according to Stirling's formula
\begin{align*}
\binom{2k}{k}\sim \frac{1}{\sqrt{\pi k}}
\end{align*}
it can be shown that for fixed $0<x<1$ and $n$ sufficiently large
\begin{align*}
\sum_{k<xn}\alpha_{2k,2n}\approx \frac{2}{\pi}\arcsin \sqrt{x}
\end{align*}
A consequence of the Arc since law are the following examples
Suppose that a great many coin-tossing games are conducted simultaneously at the rate of one per second, day and night, for a whole year.
On the average, in one out of ten games the last equalization will occur before $9$ days have passed, and the lead will not change during the following 356 days.
In one out of twenty cases the last equalization takes place within $2\frac{1}{2}$ days,
and in one out of a hundred cases it
occurs within the first $2$ hours and $10$ minutes.
(W. Feller): Anyhow, it stands to reason that if even the simple coin-tossing game leads to paradoxical results that contradict our intuition, the latter cannot serve as a reliable guide in more complicated situations.
