The article on biased coins has generated more than the usual responses. In particular, some readers seem to confuse the average value of a string of coin tosses with randomness. A fair coin is more than just one in which heads and tails have equal probability of coming up when the coin is flipped. A variety of unfair coins can be envisioned in which the average number of heads and tails are equal, but they are unfair. For instance, assume a friend assures you he has a fair coin and flips it many times. You note that the sequence is HTHTHTHT… the average number of heads is the same as tails, but you can predict the result of the next flip, and so the coin (or flipper) is not fair. (Of course, the systematic HTHTH… can, and must, arise occasionally by purely random means, but with such a sufficiently small probability that you would be well advised to make your prediction based on the pattern continuing.)

For a coin to be unbiased in the usual sense, the average number of heads must equal the number of tails, and there must be no systematic pattern which would permit an analyst to predict the next throw with better than 50% probability.

This difference is important because we rarely encounter unbiased systems in this sense when dealing with anything except games. For instance, one can argue that the prime function of the futures commodity market is to reduce the systematics in prices by allowing traders to make money predicting the future prices of commodities. The act of purchasing from intelligence in a market tends to drive prices such that the systematics are random. In an ideal market, residual fluctuations would be truly random.

To put it another way, if you can track a trend in stock prices, then the stock prices are not random. If you and enough other people see exactly the same trends and attempt to make money by utilizing them in trading decisions, the resulting trades will act to reduce the systematics. But, and this is a big but, that assumes a perfect market, which assumes complete and instantaneous distribution of all relevant information. If large players such as retirement and mutual funds all use similar automatic trading programs, they can induce artificial fluctuations due to the phase differences in information. This reality can be seen on special days such as the “triple witching day” on the stock market. Since this type of instability is recognized, the government tries to limit the growth of fluctuations and promote true randomness in the sense in which I am using that word here. That is, the price of stocks will ideally fluctuate with parameters which are known to everyone and depend on actual physical events.

We all think we understand the concept of randomness, but when we dig into it, we find it is more mysterious than it seems at first blush. The problem is that most people do not bother to dig into it. For instance, the price of homes is often given as the median price. This is probably a good way to report a meaningful parameter, but not many home buyers will be able to express why the median is a better measure than the mean, which is the type of average they are more likely familiar with. For some purposes, reporting the mode of prices would be more meaningful.

(Exercise: under what conditions would the median, mean, and mode have the same value?)

All of this occasionally highly controversial mishmash is related indirectly to the biased coin problem. If you understand the biased coin, you have a big leg up on the other, less obvious problems.

For those who wish to delve further into decision theory without wading through a lot of equations, I have posted a tutorial on elementary decision theory. It shows examples of faulty physicians’ diagnoses (important for those considering surgery) and how to evaluate anti-terrorist activities (important for everyone). That tutorial can be found here.