Just because there are an infinite number of choices doesn't make the probability of each choice equal to zero.Since there are an infinity of positive integers, the probability that you will guess it is equal to 0.
The third axiom of probability states that given an infinite sequence of mutually exclusive events A1, A2, ....
we have P(A1 U A2 U ...) = P(A1) + P(A2) + ... (The probability of their union is equal to the sum of their probabilities)
So take the following problem. I have a positive integer in my head and you have one try to guess it.
Since there are an infinity of positive integers, the probability that you will guess it is equal to 0.
Now,
1 = P(S) = P({ 1, 2, ... }) = P({1} U {2} U ...) = P({1}) + P({2}) + ... = 0 + 0 + ... = 0
What is wrong with this reasonning?
How can it be fixed?
Just because there are an infinite number of choices doesn't make the probability of each choice equal to zero.Since there are an infinity of positive integers, the probability that you will guess it is equal to 0.
I don't have emotions and sometimes that makes me very sad.
But if each number n as the same probability p then it has to.
Otherwise we get P({n}) = p > 0 for all n and then P(S) = P({1}) + P({2}) + ... = p + p + ... goes to infinity.
Of course here, I'm assuming that there is no limit on the size of the integer that someone can imagine. It is an ideal experiment.
Who said each number had the same probability? You can easily make distributions over an infinite support.
You never specified the distribution you want. If you wanted a uniform distribution over the integers then you are correct in that that distribution doesn't exist.
But your original problem isn't well defined.
I don't have emotions and sometimes that makes me very sad.
Weiss (01-31-2017)
Thanks for taking time to reply. I agree with you that the problem is not well defined.
Just to clarify my thinking.
Let's say you have to guess an integer between 1 and 10. Then the probability of guessing it is 0.1.
Now, if instead the integer is between 1 and 100, the probability will be 0.01.
If we let the upper bound of the integers increase indefinitely then the probability will be approaching 0.
So if you have to guess a positive integer from the set {1, 2, 3, ... } the probability of getting it should be equal 0 (What else could it be?).
I can see that it creates a problem since we must also have P(S) = 1. But I can not see what`s wrong in my reasoning!
You're making the unstated assumption that both the person choosing the number and the person guessing are using uniform distributions. There isn't a way to have a uniform distribution over the positive integers (as you're figuring out). That doesn't mean that it's impossible to place a guess - you just have to use a distribution other than "uniform over the positive integers" to make your guess. Now the probability that you guess correctly depends on the distribution you use to guess AND the distribution used to choose the number.
The thing that is wrong with your reasoning is that you're automatically assuming that "uniform over the positive integers" is either the only choice or is the implicit choice when posed with this problem. As stated the problem is not well defined because it doesn't specify the distributions of interest - but that doesn't mean the answer is to choose a distribution that doesn't actually exist.
I don't have emotions and sometimes that makes me very sad.
Weiss (01-31-2017)
Thank you so much! It's all clarified now.
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