I'm currently reading a book entitled "An Introduction to Probability and Inductive Logic" (it's actually a philosphy book, although it contains some calculations) and have come across something I don't understand. I was originally intending to post this in a maths forum, but it's really more of a logical problem and since you also have a background in statistics I thought you might be able to shed some light on it. I came across this forum by googling "logic forum" - I'm glad I found it because you have a lot of material on your main site which I haven't looked at in great detail yet but what I've seen so far is fascinating.
Anyway, back to the problem. In the book there is a chapter called "The Gambler's Fallacy", which is explains how and why the fallacy occurs. I thought I understood, but after each chapter there are a number of questions to test your understanding, and one of these (or rather, the answer to it) is what's puzzling me. The question is:
"Fallacious gambler strikes back.
I've been reading this old book, A Treatise on Probability, published in 1921 by famous economist John Maynard Keynes. He says that a German psychologist, Dr Marbe, studied 80,000 spins of roulette at Monte Carlo, and found that longs runs do not occur nearly as often as theory predicts. A run of 13 blacks is more improbable than anyone believes, so having seen 12 blacks in a row, I am going to bet red! (a) Is our friend still a fallacious gambler? (b) How would you explain Dr. Marbe's data?"
In the back of the book the answers are given. The answer to (a) is:
"No. Fallacious gambler sensibly supposes that "thirteen blacks" counts as a "long run", and concludes that, according to Dr Marbe's data, a sequence of 12 blacks followed by a red occurs more often than a sequence of 12 blacks followed by a black. Of course when we dig up the records, it may turn out that Marbe was referring to much longer runs, but at any rate Fallacious gambler was not committing the gambler's fallacy."
Now, I'm unhappy about this answer, because I thought the gambler's fallacy depended on the fact that you are dealing with independent trials, and you commit the fallacy by not taking this into account. Since roulette IS a game of independent trials, why is it NOT committing the gambler's fallacy to suppose that red is more likely to follow black after 12 blacks, simply on the basis of a finite data sample?
I hope you can ease my confusion. Looking forward to your answer. Many thanks,
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Ok, thanks anyway Avi. I did wonder whether it was a bit specialized.
To broaden the topic somewhat, are you of the opinion that probability is a branch of logic? I've been looking at your book "Future Logic" and it seems to me that Modal logic is closely related to the concept of probability in that it deals with the logic of possibility. I like the notion that logic cannot be clearly divided into "deductive" and "inductive", but that there is continuum between the two, with traditional deduction representing the limiting (extreme cases) at either end of the spectrum.
There seems to be a lot of controversy and debate about the interpretations of probability, even though the mathematics remains the same in every case. It seems that philosophers have been debating these various interpretations for decades, and there is still no consensus.
As a large part of "Future Logic" deals with scientific method, and must therefore be concerned with some kind of probabilistic reasoning, I would be interested to see where your methods "fit" in the current debates (if they do at all).
Perhaps I should read the book first.
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Hi Mike. Yes, I agree. probability is central to modality. The important thing to grasp here, which is perhaps the source of disagreements, is that there are different modes or types of modality - notably, natural, temporal, extensional as well as logical modality. Many logicians/philosophers focus solely on logical modality (de dicta), whereas in fact, in everyday discourse among laymen and scientists, the other modes (de re) are equally important. Yet , these different modes of modality behave differently to some extent, and have therefore to be studied separately. Each gives rise to a distinct meaning of probability.
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