Random walks and the arcsine law

Suppose you stand at 0 and flip a fair coin. If the coin comes up heads, you take a step to the right. Otherwise you take a step to the left. How much of the time will you spend to the right of where you started?

As the number of steps N goes to infinity, the probability that the proportion of your time in positive territory is less than x approaches 2 arcsin(√x)/π. The arcsine term gives this rule its name, the arcsine law.

Here’s a little Python script to illustrate the arcsine law.

import random
from numpy import arcsin, pi, sqrt

def step():
    u = random.random()
    return 1 if u < 0.5 else -1

M = 1000 # outer loop
N = 1000 # inner loop

x = 0.3 # Use any 0 < x < 1 you'd like. 
outer_count = 0 
for _ in range(M): 
    n = 0 
    position= 0 
    inner_count = 0 
    for __ in range(N): 
        position += step() 
    if position > 0:
        inner_count += 1
    if inner_count/N < x:
        outer_count += 1

print (outer_count/M)
print (2*arcsin(sqrt(x))/pi)