# The counter-intuitive of random walks accumulation Rule of the game: A fair coin is tossed $n$ times: $X_1,..,X_n$ where $X_i=\begin{cases}+1&, \text{head}\\ -1&,\text{tail}\end{cases}$ Let $S_n=X_1+\cdots+X_n$ player 1 wins if $S_n>0$ player 2 wins if $S_n<0$ It is expected that during the game, about half of the time player 1 will win and half player 2. But this intuition is wrong. Most of the time only one of the player wins most of the time. It take long time for this state to change during one game. This is because $S_n$ distributed according to arcsin and not uniformly. However, if the score is reseted and the game is repeated then both of the player will win half of the games. ## Why this happens? Define $T_n=\#\{k\le n : S_k>0\}$ i.e., the number of times player 1 wins after $n$ tosses. We will show that $P\big(\frac{T_n}{n}\le x\big)\rightarrow\frac{2}{\pi}\arcsin(\sqrt x)$ . This is called the arcsin law. The probability to win half of the games is $P\big(0.4\le \frac{T_n}{n}\le 0.6\big)\approx 0.128$ while the probability to win or loose most of the time is $P\big(\frac{T_n}{n}\le 0.1\text{ or } \frac{T_n}{n}\ge 0.9\big)\approx 0.41$. Reference:[[@What is Random? Chance and Order in Mathematics and Life]] ## Created 2026-06-12 13:32