Why odds tend to be 9 to 1 ?
Two gamblers betting with an infinity supply of cards which has two consecutive numbers n and n+1 with total number of 10^n. The betting rule is the people who sees the lower number win. A and B as participates bet each with bookie. Since both of them think the probability of their win would be 10 times more probable than their counterpart, the odds tend to be 9 to 1. I deeply doubt the conclusion and can't see why.
I don't think you've explained the game well enough to get a correct answer. What does it mean "with total number of 10^n"? Is that the same n as the "two consecutive numbers"? What is being totalled, if there is an infinite supply of cards?
There's a fundamental problem with saying "an infinite supply of cards" because there's no uniform distribution on the natural numbers. So if the paradox relies upon the gamblers comparing the finite range below them with the infinite range of higher numbers, that's a problem.
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."