One of the most commonly accepted yet fundamentally flawed ideas in probability theory is that of a fair coin. A fair coin is defined as a coin when when tossed sufficient number of times will return roughly the same amount of heads and tails. The trick here is that both the number of trials required to establish the fairness and the accuracy is left open ended.

### The evil coin

Let us say that we define some number “n” as the number of trials for which the deviation between heads and tails is “ε“. The first problem is that past performance is not indicative of future performance. Let us that the actual number of trials needed to establish the fairness of a coin in “m” where m >> n.  In such a case, the permitted deviation becomes ε/n*m. It is perfectly possible that in all the remaining m-n trials, we exactly get heads or tails and we just happened to look at a small enough subsequence of the trials when we made our initial conclusion. In effect, an evil coin that wishes to deceive us can always do so. So we can never really comment on the fairness of coin without making the assumption that what we have seen thus far is not an aberration of the true nature of the coin.

### Independent trials

Another interesting problem is one of independent trials.  If what is deemed as a fair coin has been returning a large number of heads than tails in recent trial, then probability of the next trial returning a tail has to go up every time it returns a head from this point onwards until the number of heads and tails roughly match. Either that, or we are dealing with an unfair coin. I first encountered this problem about a couple of years ago. As one of the comments point out, we are dealing with an independent and identically distributed trial problem. In the short run, getting a heads in a certain trial using a fair coin does not imply that it has increased the odds of getting tails in the next trial but as mentioned earlier, a sufficiently large trial sequences that indicates a bias towards one direction does have some implications on future trials

### Subset v/s subsequence

While there seems to be some expectation over fairness being exhibited over a subsequence of trials, the question to ask is if it can be expected over an arbitrary subset of trials from a universal set where we can observe that the coin is indeed fair. For simplicity let us restrict ourselves to both universal sets and subsets whose cardinality is even. It becomes fairly obvious that an arbitrary subset can result in an arbitrary conclusion. What can however be said is that for every subset that is biased a certain way, there is another subset that is biased exactly the other way. In effect, they cancel each other out.

A subsequence happens to be just one arbitrary subset. If so, then why do we expect to exhibit fairness? The answer lies in the subset that we did not choose. Given that we have fixed the size of a universal set and deemed it to exhibit a 50:50 distribution, we can pick any subset of choice but by doing so, we have defined the outcome remaining trials to the extent of what their distribution is going to be.

### Conclusion

All troubles seem to originate from the assumption that the coin can demonstrate fairness over the next “n” trials. That it will actually do so is not guaranteed and more importantly, if it doesn’t do so it means nothing i.e. statements like “it has a 99% chance of doing so” means nothing since you tend to reinvent the problem by now making the unit of your trial as the original “n” trails and start all over again.