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)

It’s a little Python 3 script. 😉

A note: The program has an error: it always prints zero for the MC result due to integer division. (I use python 2.7). This mod fixes it.

print (float(outer_count)/M)

Cheers, dan

Very nice.

By the way, there is a small bug in your new design. Your code snippets have a small square at the top right hand corner with a z-index higher than your header. As a consequence of this, when you scroll down the square overlaps your header.