Apatriarca defined a function P that maps some subsets of the natural numbers to real numbers between 0 and 1. This function is not a probability, as it does not satisfy the third axiom described here. If you relax the axiom to require only finite sums to work, then it would be fine.

Alternatively, we could forget about natural numbers and work with infinite sequences of bits. Or, almost equivalently, we could imagine there is a "0." before the sequence of bits, and what we are doing is picking a random real number between 0 and 1 uniformly (technically we are assigning probabilities to subsets of [0,1] using the Lebesgue measure, for the sigma-algebra of measurable sets). Now "the first bit is set" means "x >= 0.5" and the "first two bits are set" means "x>=0.75". Computing those kinds of probabilities is easy, since the probability of an interval is its length.

Now, what was the question again?

if you know the low bit is set, that eliminates half of them.

if you know the second from lowest bit is also set, that eliminates half of those remaining.

the problem is you have an infinite number, so half or 1/4 of them is still an infinite number.

only when you have a finite number (say 1000) does calculating odds make sense.

if you have 1000, and you pick one, your chances of picking the right one are 1 in 1000.

if you know the low bit is set, you now only have 500 to choose from and the odds drop to 1 in 500.

if you know the two low bits are set, you now have 250 to choose from and the odds are 1 in 250.

The OP is quite unable to express him meaningly, But this tricks me too, I will elaborate further.

Someone picks any positive whole number (N={0,1,2.....}).

- what is the probability I pick the same one? (it cannot be 0 since it is possible to happen, but extremly unlikely to happen )

- if someone says the number is odd, how bigger is probability I will gess what number is picked?

We are speaking of a number from N set. (I do not know how those numbers are called in english exactly,I am talking of the set N={0,1,2,...})

Interisting question becouse of wheather disputing meaning of them, the question just makes sence.

The probability being a real number : f e R; f e (0,0,1.0) ; f>0.0 ;f<1.0;

There is no such thing as a uniformly random natural number. If someone picks a natural number randomly, he must do it with some probability distribution (the probability of the singletons {n} forms a sequence of non-negative numbers whose partial sum converges to 1). Then you can answer any question you want.

Read my previous post (#21) for more details.