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Probability

I recently went to gamble at a casino. First timer that I was, I decided to stick with the roulette table. In particular, I decided to only bet on black and red (ALMOST 50% chance of success, as one must also factor the number 0, which is neither red nor black). After a while observing the results board (and not betting), I noticed that what appered to be a chaotic pattern of results, became a pretty steady and predictable 3-reds-3-blacks type of pattern. So I began betting, and, lo and behold, I began winning. Naturally, every now and then I would lose some (sometimes there would be 4 blacks in a row instead of 3)--but, overall, I was winning. Then this magical pattern vanished, giving way to the same chaotic (that is: to my eyes) pattern which I had observed at the beginning. After, say, 1 hour, the 3b-3r pattern was back in place. Days later, I returned to the same casino, and didn't even place one bet: my magical pattern just never manifested itself! Any explanations? I am no mathematician, but...

I'm no expert on probability, but I think you have to consider different types of pattern differently. In the case of the coin, a pattern of 50 heads and 50 tails (in any order) is much more likely than 99 heads and 1 tail. But a pattern of first 50 heads and then 50 tails is no more or less likely than a pattern that alternates heads and tails, or one that alternatives pairs of heads and tails, etc. Similarly, if you deal out 13 cards from a well-shuffled deck, you are very unlikely to get all 13 hearts, but that outcome is no less likely than any other specified hand.
Probability

How popular is Bayes' theorem among philosophers? As a physicist, it has had a profound effect on my thinking, and seems to reflect the way we intuitively deal with new evidence presented to us. As a reminder, Bayes' theorem states: Probability(A given B) = Probability(B given A)*Probability(A)/Probability(B) For example, if A is "A revolutionary new theory" and B is "Data from my experiment", then Bayes' theorem tells us that we have to take into account our initial (prior) belief in the theory P(A), given our background knowledge, before even looking at our data.

Bayes' Theorem is very popular among philosophers of science who work on the bearing of evidence on theory. As you say, it has some attractive features. In your formulation, "A" stands for the theory and "E" for the evidence. To keep this straight, I'm going to use "T" and "E" respectively. If we take the P(T given E) to be the probability that theory T has after you observe evidence E and P(T) the probability the theory had before, then the difference between these is naturally taken to be the degree to which the evidence supports the theory, and Bayes theorem plausibly says that this will be greater the greater the probability of P(E given T) -- where this probability peaks at unity if T entails E -- and the smaller P(E), the probability of the evidence before you observed it. In other words, this take on Bayes theorem says that you get the strongest support from surprising evidence which would however have to be true if your theory is true. And that sounds right. Of course Bayesianism...