coins.py

"""
A program for investigating this property of coins:
If Alice tosses a coin until she sees a head followed by a tail, and Bob tosses a coin until he sees two heads in a row,
then on average, Alice will require four tosses while Bob will require six tosses (try this at home!),
even though head-tail and head-head have an equal chance of appearing after two coin tosses.
From https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/
"""

from numpy.random import randint
from sys import argv

HEAD = 0
TAIL = 1

# Give N on command line.
N = int(argv[1])

"""
Flip a coin. 0 is heads, 1 is tails.
"""
def throw_die():
    return randint(2)

"""
Returns the total number of coin tosses needed
to satisfy the conditions given. It is possible to give any
combination for two consecutive throws:
Head and then tails, head and then head, tails and then head,
or tails and then tails.
"""
def experiment(N, first_throw, second_throw):
    total = 0
    for i in range(N):
        old = throw_die()
        throws = 1
        while True:
            new = throw_die()
            throws += 1
            if old == first_throw and new == second_throw:
                break
            old = new
        total += throws
    return total

total_Alice = experiment(N, HEAD, TAIL)
total_Bob   = experiment(N, HEAD, HEAD)

print "With %i experiments," % N
print "%10s %5.2f" % ("Alice:", total_Alice/float(N))
print "%10s %5.2f" % ("Bob:", total_Bob/float(N))