TITLE: What is the probability you guess the number I am thinking of?
QUESTION [6 upvotes]: Probability is defined as the likely number of outcomes over all total outcomes.  In this case, 1 over infinity; which would equate to zero.
But, there is a chance you can guess the number I am thinking of.
Furthermore, would it depend on the number of guesses you have?
Let's say you have 2 guesses? 100 guesses? Infinity guesses?
I'm looking for any insight as to the answer to this problem whatsoever.  Is there even a way to answer this question?
Let me know.
Thanks!

REPLY [2 votes]: I suspect that you are getting more than you asked for in some of the answers.  The key is in the third paragraph of @julien's answer: for the probability of picking each number to be finite (greater than zero, informally), the probabilities must be distributed in a way that gives a finite sum.
So, you cannot give an "equal" weight to "every" number, because the weights sum to infinity and cannot be normalized.  However, there are plenty of distributions which do work, and if we are talking about a real human choosing a number, it seems valid to assume that very large numbers are unlikely to be chosen.
The simplest example would be to weight the integers to have probability $\frac1{2^n}$.  Thus 1 has probability $\frac12$ of being chosen, 2 has $\frac14$, and so on; this sums to one for the total probability of a positive integer being chosen, but every number has a finite probability.
Just to show there are other possibilities, we could say that the probabability of choosing a d-digit integer was $\frac9{10^d}$, giving 1-digit numbers a 90% chance in total, 2-digit numbers 9%, 3-digit .9%, etc.  Again, the total probability sums to one but the individual probabilities are all finite: 10% each for 1 through 9, .1% each for 10 through 99, etc.
So, it all comes down to the assumed distribution of chosen numbers, which ultimately depends on a model of human thinking.