In my very limited mathematical knowledge, at first the answer seems to be as N →∞ 1/N which we know equals zero. But, what if the number is 7, and that is what was picked?
do you mean mentally? "I'm thinking of a number." or do you mean out of a drum, etc. If it's a physical draw you have the answer, if it is a mental game then the number chosen is the finite set, and then becomes a chicken and egg problem. Which came first the finite set or the number chosen that created the finite set.
If you want to talk about a probability of doing something, then you need to have a probability distribution. So, what is the distribution you're using? Also, probability 0 doesn't mean that something is impossible.
Are you saying that there are different kinds of 0 probabilities? Is an impossible probability like saying: what is the probability of 6 being the right number out of numbers 1 to 5?
Saying that the natural numbers have a uniform distribution from 1 to infinity doesn't even really make sense. Last time I checked there is no uniform distribution on the natural numbers. Edit: Yes probability zero doesn't have to mean it won't ever happen. In the case you described it does mean that because the sample space you described does not contain the element you picked, thus if you put the numbers 1, 2 , 3 ,4 ,5 and a hat and picked a random number, then you know you won't make a 0 or 10 or 10000, and if you do, then you know your sample space was wrong. On the other hand, without getting into some more technical language, if a sample space is uncountable then a zero probability is almost always going to happen in many probabilistic models.
It only makes sense to talk about the probability of an event in the sample space. If your sample space consists of the numbers 1 through 5, the probability of drawing 6 isn't zero; it isn't anything at all. A probability of 0 doesn't imply that an outcome is impossible: The probability of drawing any specific number from a normal distribution is 0, but clearly you must draw something.
Your error is trying to extend concepts from finite, discrete, probablity spaces to infinite spaces. In finite probability spaces, a probablity of "0" can be interpreted as "impossible". Not if you are talking about infinite spaces. (And, similarly, in infinite spaces, a probability of "1" does NOT mean "certain to happen".)
Probability 0 means that it sure that the event "will not" happen, e.g. like probability of getting a red ball from bag of blue and green balls. Probability 1 means that it is sure that event "will" happen, e.g. like probability of getting 2 red balls from bag of 2 red balls But what your question is little strange, because probability cannot be taken from indefinite sets there has to be a finite set of reference. Are you understanding this........... PM me for more help!
This is an interesting question more philosophical perhaps. So, what is the probability of any number? well it would have to be 0 based on the limit lim N→∞ 1/N. Like what is the probability I would say a number never in all history been said or written? Well it would still be 0. But, the probability must be increasing if I exclude all numbers been said. So first it is probability of zero, but it's increasing. But, infinite is what I like to call the a dynamic number, so even when you remove a number it has a replacement in the sense that it's infinite. Any way you look at it. It is most reasonable to conclude a probability of 0 does not mean it's impossible, but then what does a probability of 1 mean? .... This is where the universe ends.
Do YOU understand that this is NOT always true? And every answer here has been saying that. If you have a "uniform" probability distribution for all numbers from 0 to 1, the probability that you will choose any number from 0 to 1 is 0. But some number has to be chosen!
As mentioned above we cannot make a uniform distribution of the natural numbers. We can define a nonuniform distribution or a uniform distribution of a function. For example we can give a sensible definition that allows us to say the probability that a uniform random integer is even is 1/2.