Skip to main content

Python: Getting a Fair Flip out of an Unfair Coin

If you have an unfair coin (i.e. one that favors heads or tails), how do generate a fair flip (i.e. one that doesn't favor heads or tails)? My buddy Hy Carrinski and I came up with the following algorithm:
"""Get a fair flip out of an unfair coin."""

from collections import defaultdict
from random import random


def flip_unfair_coin():
"""Flip an unfair coin. Return a boolean."""
return random() > FLIP_RATIO

def create_fair_flip():
"""Generate a fair flip. Return a boolean."""
while True:
flip = flip_unfair_coin()
if flip != flip_unfair_coin():
return flip

# Demonstrate that it works.

if __name__ == '__main__':
results = defaultdict(int)
for i in xrange(1000000):
results[create_fair_flip()] += 1
percentage = (results[False] / float(results[False] + results[True])) * 100
print "Percentage heads:", percentage


Dirk Bergstrom said…
It works, but I don't really understand why (that's what happens when you don't learn statistics). An explanation would be nice...

While we're at it, there's a typo in the script: you write "Prove" but you merely demonstrate.
It works because even on an unfair coin, the chance of a heads followed by a tails is the same as the chance of a tails followed by a head. This algorithm only returns a value when one result is followed by the other. Looking at the possible outcomes makes this pretty clear. If heads is .7 and tails is .3, then the four possible outcome probabilities are:

heads/heads = 0.49
heads/tails = 0.21
tails/heads = 0.21
tails/tails = 0.09

The algorithm throws away heads/heads and tails/tails and returns tails for heads/tails and heads for tails/heads - each with a probability of of 0.21 per attempt (with transparent re-tries for the rejected cases).
Luke Plant said…
Very neat!

@Dirk: it works because even biased coin flips are still independent. (if they are not e.g. if a person is controlling the flips, or the coin has been set to do a certain sequence, this method will fail).

Essentially, you are flipping the coin twice, and then only looking at the times when the two results are different. The first time you get that scenario, pick the first result (or last, it doesn't matter). The probability of getting Heads then Tails is the same as the probability of getting Tails then Heads (due to independence), so you get odds of 50% for the overall result being Heads or Tails.
Jeff Epler said…
You've reinvented the von Neumann Extractor, and Jean-Paul is right about the reasons it works for a biased but non-autocorrelated source.
jjinux said…
> While we're at it, there's a typo in the script: you write "Prove" but you merely demonstrate.

Updated. Thanks.
jjinux said…
> It works because...

Wow, great explanation!
jjinux said…
> You've reinvented the von Neumann Extractor, and Jean-Paul is right about the reasons it works for a biased but non-autocorrelated source.

Nice job providing the reference. I'm 100% okay with the fact that I came up with the same thing as Von Neumann ;)
Paddy3118 said…
We have a task on this very topic here:, so you can see it solved in Python and other languages.

Popular posts from this blog

Drawing Sierpinski's Triangle in Minecraft Using Python

In his keynote at PyCon, Eben Upton, the Executive Director of the Rasberry Pi Foundation, mentioned that not only has Minecraft been ported to the Rasberry Pi, but you can even control it with Python. Since four of my kids are avid Minecraft fans, I figured this might be a good time to teach them to program using Python. So I started yesterday with the goal of programming something cool for Minecraft and then showing it off at the San Francisco Python Meetup in the evening.

The first problem that I faced was that I didn't have a Rasberry Pi. You can't hack Minecraft by just installing the Minecraft client. Speaking of which, I didn't have the Minecraft client installed either ;) My kids always play it on their Nexus 7s. I found an open source Minecraft server called Bukkit that "provides the means to extend the popular Minecraft multiplayer server." Then I found a plugin called RaspberryJuice that implements a subset of the Minecraft Pi modding API for Bukkit s…

Apple: iPad and Emacs

Someone asked my boss's buddy Art Medlar if he was going to buy an iPad. He said, "I figure as soon as it runs Emacs, that will be the sign to buy." I think he was just trying to be funny, but his statement is actually fairly profound.

It's well known that submitting iPhone and iPad applications for sale on Apple's store is a huge pain--even if they're free and open source. Apple is acting as a gatekeeper for what is and isn't allowed on your device. I heard that Apple would never allow a scripting language to be installed on your iPad because it would allow end users to run code that they hadn't verified. (I don't have a reference for this, but if you do, please post it below.) Emacs is mostly written in Emacs Lisp. Per Apple's policy, I don't think it'll ever be possible to run Emacs on the iPad.

Emacs was written by Richard Stallman, and it practically defines the Free Software movement (in a manner of speaking at least). Stal…

ERNOS: Erlang Networked Operating System

I've been reading Dreaming in Code lately, and I really like it. If you're not a dreamer, you may safely skip the rest of this post ;)

In Chapter 10, "Engineers and Artists", Alan Kay, John Backus, and Jaron Lanier really got me thinking. I've also been thinking a lot about Minix 3, Erlang, and the original Lisp machine. The ideas are beginning to synthesize into something cohesive--more than just the sum of their parts.

Now, I'm sure that many of these ideas have already been envisioned within, LLVM, Microsoft's Singularity project, or in some other place that I haven't managed to discover or fully read, but I'm going to blog them anyway.

Rather than wax philosophical, let me just dump out some ideas:Start with Minix 3. It's a new microkernel, and it's meant for real use, unlike the original Minix. "This new OS is extremely small, with the part that runs in kernel mode under 4000 lines of executable code." I bet it&…