Is Memory In The Cards?

A friend of mine is going to Las Vegas with high hopes of being a net winner this trip. The last couple of time he went there, he ended up a net loser, so he is convinced that by the law of averages, he is due to win. I asked him if the cards have a memory. He asked what I meant.

His mistake is a common one. If we know that the odds of winning some game such as betting on a coin toss are 50/50, and we have been losing, then we should be due for a winning streak. Sounds good. But how does the coin know what your score is?

This common error is sometimes defended by a misapplication of the principle of “Regression to the Mean,” Taken at face value, this well-established statistical principle seems to say that if you gamble a lot with known odds, that a losing streak will be cancelled out by a winning streak. (Technical note: the original name for this phenomena was “Regression to Mediocrity” by Galton  http://www.mugu.com/galton/), but we now have a different meaning of “mediocrity” than was common in Victorian England, so the name of the phenomena was changed to match cultural norms.)

While it is true that in the long run, other things being equal, your actual total rate of return will approach closer and closer to the theoretical return given the rules and your skill, that doesn’t say anything about how you will do on the next toss of the dice or deal of cards. Neither dice nor cards have memories.

To put it another way, when do you start keeping your score? Suppose you keep your score starting on day one at Las Vegas and have a losing streak. Your companion of choice notices this and starts to keep score on the second day just as you go into a winning streak. If you play for many more days, when you compare notes, both people should show about the same rate of return on investment (or, rather, rate of loss on investment). The odds in the long run are not determined by when you started counting.

So what is regression to the mean and why is it important? First, it is a real thing. Second, it is easy to misapply it.

Rather than trying to give the full meaning here, I did the obvious and asked Google. If you do that, many interesting sites turn up. I reviewed several of them and recommend this one http://www-users.york.ac.uk/~mb55/talks/… as being more lucid than many. Skim through the first couple of paragraphs and dig into the graphs of two people taking pulses of volunteers. However, this one http://www.ruf.rice.edu/~lane/stat_sim/r… popped up first and has a nifty Java applet - but it is more difficult to understand for a first time exposure. It makes more sense if you already have an idea of the underlying concept.

The conditions under which a regression to the mean can be correctly applied are strict, but not complex. But be careful about hidden assumptions and remember - dice and cards don’t have a memory. They have no idea of regression. If you have a long string of losses and think you are ready for some wins, I’m sorry to report that the very next deal or toss will have the same odds as before.

Of course, if you are gambling on a coin toss by another person and it comes up heads 20 times in a row, and you have consistently bet on tails, what would you bet on the next toss? The unbiased odds of 20 heads in a row is about one in a million. The unbiased odds that the person tossing the coin is cheating are probably much higher. And if that person is cheating, the cheat can likely go either way. Therefore if you switch to betting on heads, do not be surprised if the next toss comes up tails. But this is not due to regression to the mean or to breaking a losing streak. You have more likely been cheated.

And that is one of the strict conditions of regression to the mean. Correct application means that the underlying paradigm is constant. That is, throughout the measurements (gambling, etc.) the system remains constant. Manipulation of the odds by a factor outside the system changes the results.

In the meanwhile, cards do not remember (have I said that already?).

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.