What you see in the casino is the statistical laws of probability being expressed as a sequence according to the laws of random distribution. This can impact on the distribution of race results to a limited extent, but as Gareth points out though, in racing there is a 'cause and affect' dynamic at play, and that is why sequences can be explained in nearly every instance (admittedly with differing degrees of certainty) and with a degree of speculative hypothesising in the event of a new one emerging until it becomes more or less accepted or disproven. The secret therefore is to establish whether there's a rational, and importantly, satisfactory (the two aren't necessarily the same) explanation for a sequence. Once you understand the dynamic involved, and are happy that it's more than just random distribution, then you can use the trend to inform your judgement, based on an appreciation of it's relative reliability as an indicator (its rare to find a 100% filter, without getting silly).
The problem is when an unexplainable sequence occurs which seems to have no tangible cause and effect but none the less, appears to be a demonstrable trend of reliable forecasting ability. For instance what stock would you place in a correlation between the annual percentage conception rate in Argyllshire for the last 100 years amongst the population, being the same as the winning stall number for the Derby? These types of instances are rare to unheard of (especially on this scale) and have to be chalked up to conicidence, (though with a correlation that strong, you'd be tempted to have a few shekels I'd suggest?)
There's also a pyschological element to some trends/ conicidences also, which involves people finding what they want to find. Mordin's often accused of it, but one of the best examples I can think of concerns the whole list of coincidences that are supposed to link the assasinations of Abraham Lincoln and Jack Kennedy. When you look at the resevoir of data available for any events in recorded history (linked or otherwise) and how you can then extrapolate it for interpretation, its no surprise that people can generate the seemingly 'weird'. This is little more than selective coincidence building, and totally consistant with random distribution given the numbers involved.
I think what you're describing Euro, is equally dangerous though, and a trap that people can fall into (I know I could). It's a situation built around confusion essentially, a sequence that defies probability can be generated, and presented as something above random distribution. In theory it should self regulate and correct itself over time, returning to a more predictable pattern. Failure to do this however, will mark it out as a shift in a trend that requires explanation. It depends where abouts you are on any hypothetical schedule therefore as to how you interpret it. Climate change would be a good example for instance. Do you say that the 4 hottest years in recorded global history occurred in the 1990's (its probably worse by now) but we needn't worry, it's a blip? I think it's easy to confuse a sequence and a trend though. One is tantamount to a series of results, the other involves an explanation of this sequence, and therefore carries an implicity element of future forecasting.
