Even if something has probability 0, it can still happen.
For example take your body length. What is the probability of anyone having exactly that length? Yet you have it
A person is playing a game where they guess a number between 0 and infinity. Is it correct that the person has no chance in guessing the right number because ? But surely the chances are better than not guessing at all so shouldn't it be greater than 0?
Part of the problem is knowing what is meant by asking someone to "guess a number between 0 and infinity". (Let's assume for a start that "number" means "positive integer" here.) If there was an equal probability of any number being chosen, then that probability would have to be zero. As Mobius points out, having probability zero does not mean the same as being impossible. But in fact it is not the case that there is an equal probability of any number being chosen. If you ask someone to choose a number, they are much more likely to choose 7 say, or maybe 89, than they are to choose 4967456195060002851693452 for example.
In fact, it would not be practicable to design a random number generator that would produce any number with equal probability. The first time you asked it to give you a number, it would almost surely start listing a number with so many digits as to outlast the lifetime of the machine (or the observer).
Isn't this merely a practical limitation, mostly caused by the fact that humans (or who/whatever you substitute for 'someone' ) are very bad random generators?
This also seems just a practical argument to me, moreover because the same holds for a random number generator that would produce any real number between 0 and 1 with equal probability (i.e. a Uniform(0,1) distributed variable). This would most likely output a number with more decimals than fit in the universe in whatever way you'd try to represent it?In fact, it would not be practicable to design a random number generator that would produce any number with equal probability. The first time you asked it to give you a number, it would almost surely start listing a number with so many digits as to outlast the lifetime of the machine (or the observer).
If not, why couldn't you restrict the latter example to real numbers of the form 1/n and return those instead of n. I'd say if you can output one, you can output the other?
Not a practical limitation, it is in fact impossible (to generate naturals with a uniform distribution), it is in fact easy to show that the probability that such a number has digits or fewer is 0, for any . That is the probability that it has more digits than is 1.
That would be with probability 1 (that is the number would be transcendental with probability 1)This also seems just a practical argument to me, moreover because the same holds for a random number generator that would produce any real number between 0 and 1 with equal probability (i.e. a Uniform(0,1) distributed variable). This would most likely output a number with more decimals than fit in the universe in whatever way you'd try to represent it?
These don't have a uniform distribution on [0,1]If not, why couldn't you restrict the latter example to real numbers of the form 1/n and return those instead of n. I'd say if you can output one, you can output the other?
CB
So the first has "infinite" digits, the second has "infinite" decimals. ("infinite" here meaning more than any n with probability 1)
Why is that a problem for the first, but not for the second? That is assuming we consider the idea of a Uniform(0,1) distribution to be valid and possible.
I agree it would be impossible to design a random number generator that actually outputs random natural numbers. But then again I think it would be just as impossible to design one that actually outputs random real numbers between 0 and 1.
Because there are natural process which can be used to produce true random numbers from a known continuous distribution we can use the transform method to generate a true ~U(0,1) RV. But that is largely irrelevant since we cannot record a real value with arbitrary precision and we always end up with a requirement for a RV which takes a value corresponding to one of discrete set of values in the unit interval that approximates an ideal uniform distribution.
(even if we could produce random numbers ~U(0,1) we could never output them as almost all reals in the unit interval are un-nameable)
CB
Is the phrase 'almost surely' relevant here? That's to say: the person will almost surely not be able to guess the right number?