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RE: Why odds tend to be 9 to 1 ?
there are 10^n cards with n on a side and n+1 on the other side, for example, there are 10 cards with 1 on a side and 2 on other side, 1 million card with 6 on a side and 7 on the other side.
The key is to ask, what is the distribution of cards that can be drawn?
As I stated above, it cannot be uniform over all of the infinite set of cards. We can't even answer a simple question like "what is the probability that the card has a number less than X" under the assumption of uniform distribution, because it doesn't make any sense. So if you assume that "each card is equally likely" you will inevitably get a contraction because that's impossible to start with.
On the real numbers you can have a probability distribution where the value of each event is zero but the integral is nonzero, but even there you can't have a distribution over "all real numbers" One a discrete set like the natural numbers you can't make it work the same way--- there's no uniform probability you can assign that makes the probability sum to 1, as it must.
So, if we have a finite number of cards, there is no paradox, and if we adopt a nonuniform distribution over the infinite cards, then we can calculate the probability without paradox. But if we assume something that doesn't exist (a uniform distribution over card sets of size 1, 10, 100, 1000, ... without bound) then we're doomed from the start.
This is a paradox from Littlewood' Miscellany named infinity paradox which aims to provide a taste of mathematics of infinity.
This one?
As the book says, "will probably not stand up to close analysis." There is no such thing as "infinitely probable."