# 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)
```

## 3 thoughts on “Random walks and the arcsine law”

1. Steve Spicklemire

It’s a little Python 3 script. ;-)

2. dan sullivan

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

3. 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.