# Are Coin Flips Actually Unbiased?

This one has always bugged me. As subjects, math and stats tend to be a bit more unforgiving in terms of assumptions, and yet even at higher levels coins are sometimes still used in practice as a 50/50 binary random variable generators without much question from some. I looked into it, and found that I am far from being the first skeptic.

Here is a detailed paper resulting from collaboration between mathematicians on the subject. They did some very impressive experimental and theoretical work on the subject which yielded potentially surprising results.

An interesting interpretation of the paper can also be found here.

Basically, aside from existing ways to “cheat” with a coin flip through skilled technique, some hypothesize that a regular coin toss has a 51% chance of landing with the same side face up as when flipped. One possible basic idea can be understood by looking at the finite sequence of “Heads” up or “Tails” up that any flipped coin must go through.

(ex: HTHTHTHTHTHTHTHTHTHT…. where the landing position is either H or T)

By starting a coin flip with “Heads” up, you are guaranteeing that there will never be a higher proportion of “Tails positions”. At best, they will be tied. So, there is a small boost in the probability of landing “Heads”, or whatever the initial position of the coin was…Or is there? It can be debated, and although the ideas are interesting no real consensus has been reached.

The actual arguments in the paper are more physical in nature and are a little different, but are also in favor of landing on the same side up as when flipped.

Suppose we are still interested in using a coin toss to, say, allocate patients to treatments under a certain randomization scheme. Is all well and fair, so long as the coin-flipper has no technique and is blind to the initial position of the coin?

(Note: I would use simulation software for this, but I wonder if anyone actually does still flip coins).

Or should another investigator be around, and randomly decide to turn the flipped coin over (as is customary in many playgrounds)? Or maybe the coin should be caught sideways?

What about professional sports? Coin tosses play a big part in those.

Just brainstorming, but the main idea I’m exploring is that coin tosses might not deserve the innocent appreciation that they are currently enjoying, and we perhaps should all discuss it more.

Many dispute the interpretations of the results of the paper, and there are some pretty lively debates about all aspects of the study. Basically, the authors suggest that at least 250000 genuine flips would be required to detect the kind of bias they are hypothesizing, which far surpasses any world record yet set for consecutive flips.

Better solution? A well known way to get a fair flip out of an unfair coin.
Everyone should just do this if they suspect the coin they are flipping with may not be fair.

Turn an unfair coin into a fair flip algorithm:

Flip coin two times.
If it ends up HT or TH, the result is the first one of the sequence.
If it ends up HH or TT, continue flipping it again two times until you get HT or TH.

This works because if the probability of getting H is p and getting T is (1-p), we have
p(1-p) = (1-p)p, meaning that the events HT and TH are equally likely. So, by choosing one of two equally likely events(HT or TH), we have turned this into a fair game.