Kudos to NOTI for clearing up some of the misconceptions about pigeons and the Monte Carlo problem. (Seriously)
What I want to offer up is that I think Frank introduced a red herring into the general topic when he suggested that "it's a bad idea to look for patterns in random events." Hell, scientists engage in queries all the time where they do just that. SETI is one of the most notorious examples.
The fallacy that I believe Frank desired to highlight was attributing meaning to patterns in random events without first having subject the phenomenon to appropriate statistical tests to evaluate the probability of such a pattern occurring "at random", as opposed through some distortion of natural odds.
When it comes to the belief that some have that they can improve their odds in the casino by strategically switching machines, it has to be conceded that the M.C. pigeon "switchers" were right on the money. But the math fully supports their call .. intuitively, they made the right choice.
But in some situations intuition is a poor play partner, and the casino "machine switch" has no such support in the underlying math. As I noted before, ideally, such behavior has no penalty (such as when the switch is to another equivalent machine and doesn't impact other aspects of play). But, in practice, I often see it give rise to sub-optimal choices that impact the player, and sometimes those around them.
