Suppose someone described a lottery with three tickets, where it was certain that some ticket won, and the probability of each ticket winning was 1/100. We could note that the description violates additivity. Or, more intuitively, we could say that while lottery would make sense with, say, win probabilities 1/3, 1/3 and 1/3, the claimed 1/100 is just way too small a probability, because it is so much smaller than a set of probabilities that do make sense for a lottery with these many tickets.
We can say the same thing about infinite countable lotteries with infinitesimal outcome probabilities. Suppose the proposed probability of each individual ticket winning is some infinitesimal u. Now consider a perfectly probabilistically sensible and unparadoxical, albeit unfair, infinite lottery with individual win probabilities 1/2, 1/4, 1/8, 1/16, .... That lottery makes perfect sense. But our alleged infinitesimal probability lottery has the same number of tickets, but assigns to each one an infinitely smaller probability, since u is infinitely smaller than 1/2n. And so our alleged infinitesimal probability lottery assigns much too small a probability to each ticket.
4 comments:
Isn't this paradox resolved by pointing out that even though each individual element u is smaller then every individual element 1/2^n, it doesn't follow that the whole of the first series can't be the same than that of the second series.
Just as having an infinite number of 1 meter columns stacked on each other will have the same length as an infinite number of 1 centimeter columns stacked on each other.
"Just as having an infinite number of 1 meter columns stacked on each other will have the same length as an infinite number of 1 centimeter columns stacked on each other."
Yes and no. In terms of cardinality counting, yes. But in terms of non-standard analysis, where we have a particular infinity K, K meters > K centimeters. And non-standard analysis becomes salient when we are talking about infinitesimals.
This argument is also my forthcoming Synthese paper.
Preprint here.
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