On Sat, May 1, 2010 at 4:08 PM, Brent Meeker <[email protected]> wrote: > > Seems like a good answer to me. Suppose there were infinitely many rolls of > a die (which frequentist statisticians assume all the time). The fact that > the number of "1"s would be countably infinite and the number of "not-1"s > would be countably infinite would change the fact that the "not-1"s are five > times more probable.
So let's say that we have an infinitely long array of identically sized squares. Inside each square a single number is written, from 1 to 6. First let's say that the numbered squares just repeat: 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6...over and over, infinitely many times. Now, we randomly throw a dart at this infinitely long row of squares. Should we expect to hit a 1, or not-1? Not-1, right? Because we have extra information about the internal structure of the infinitely long row. The dart has to hit in some finite space, and the layout of the numbers in the squares for any given finite space is known. So the probability of hitting a "1" is "1 in 6". NOW. Let's say the ordering of the numbers in the squares is completely random. We've lost information here. When we throw the dart at the row, we have no idea what numbers will be in the randomly selected finite area we aim towards. In an infinite sequence, any given finite sequence will appear infinitely often...so there are stretches as large as you want to specify that contain only 1s or only not-1s. Further more, as you say, the 1's and not-1's can be put into a one-to-one correspondence...both sets are countably infinite. There are as many "1's" as "not-1's". And there are as many 2's as "not-2's" and so on. So, we lost a lot of information there when we abandoned the strictly repeating structure. Before we lost that information, we could safely say that the probability of hitting a 1 was "1 in 6"...but after losing that information surely we can't say anything at all about the probability of hitting a "1" with our dart. "Whereas the interpretation of quantum mechanics has only been puzzling us for ∼75 years, the interpretation of probability has been doing so for more than 300 years [16, 17]. Poincare [18] (p. 186) described probability as "an obscure instinct". In the century that has elapsed since then philosophers have worked hard to lessen the obscurity. However, the result has not been to arrive at any consensus. Instead, we have a number of competing schools (for an overview see Gillies [19], von Plato [20], Sklar [21, 22] and Guttman [23])." (http://arxiv.org/PS_cache/quant-ph/pdf/0402/0402015v1.pdf) -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [email protected]. To unsubscribe from this group, send email to [email protected]. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
