Ant on a cube probability

Probability Level 3

An ant is on one face of a cube. At every step, the ant crawls to one of its four neighboring faces with equal probability.

What is the expected number of steps for it to reach the face opposite its starting face?


The answer is 6.

Michael Wang by Michael Wang , 17, USA

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2 solutions

Fletcher Mattox
Aug 8, 2020

In lieu of an analytical solution, I'll offer 

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import random
import numpy as np

d = dict()
d['N'] = ['S', 'W', 'U', 'D'] 
d['S'] = ['N', 'E', 'U', 'D'] 
d['E'] = ['S', 'W', 'U', 'D']
d['W'] = ['N', 'E', 'U', 'D'] 
d['U'] = ['N', 'S', 'E', 'W']
d['D'] = ['N', 'S', 'E', 'W'] 

steps = []
for i in range(10**6):
    pos = 'N'
    dst = 'E'
    step = 0
    while pos != dst:
        pos = random.choice(d[pos])
        step += 1
    steps.append(step)
print(np.mean(steps))


1
 
5.99812

1 Helpful 1 Interesting 0 Brilliant 0 Confused
Yuriy Kazakov
Sep 27, 2020

Markov chain with absorbing state

Wolframalpha for Markov chain with absorbing state

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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