February 5th, 2013, 12:23 AM  #1 
Newbie Joined: Dec 2010 Posts: 8 Thanks: 0  Infinite set of numbers
If you pick n (n > 0) numbers at random from infinite set (say positive whole numbers) what is the probability you pick a specific number (1 for example)? The question bothers me because from one side it seems the probability must be greater than 0 (since some numbers are being picked), but from other side it seems that the probability is zero since the set is infinite. Thanks in advance 
February 5th, 2013, 12:52 AM  #2 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Infinite set of numbers
Such probability is not possible to calculate assumed that probability of picking every integer is the same. Say the probability you picked any integer N is P(N). Since the probability of picking every integer is the same, P(N)=k. Sum of the probabilities of picking each and every integer is 1  P(1) + P(2) + . . . = 1, hence k + k + k + . . . = 1 which is impossible for any positive real number k. Hence, the probability doesn't exists.

February 5th, 2013, 05:15 AM  #3 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Infinite set of numbers
There is no uniform distribution on a countably infinite set; your question contains an assumption which is not true.

October 18th, 2015, 10:40 AM  #4 
Newbie Joined: Oct 2015 From: India Posts: 2 Thanks: 0 
You first have to understand the meaning of infinite... Infinite in maths is like beyond boundary ... If you can think only to 99 ... 100 is infinite for you.. So it is not define what no. Are included... Probably probability only apply to definite things Bcoz it is based on tries and success.
Last edited by skipjack; October 18th, 2015 at 11:32 AM. 
October 18th, 2015, 12:29 PM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,621 Thanks: 2609 Math Focus: Mainly analysis and algebra 
Hardik's comment is false. The Poisson and Geometric distributions comfortably handle the infinite within a discrete probability space. The Normal distribution does so in a continuous space. These are only examples  there are infinitely many distributions that do the same. But any discrete distribution that handles the infinite must have $\Pr{(X=x)}\to 0$ as $x \to \infty$, and any continuous distribution must have $\Pr{(X\gt x)}\to 0$ as $x \to \infty$. This is because the cumulative probability distribution function must tend to unity. 

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