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If I had an infinite number of sticks and (somehow) painted sticks #1,4,7,10.. in red and then painted sticks #2,3,5,6,8,9... in blue.

Then I picked a stick at random, do I have more chance of picking a blue stick or is the probability the same because there are an infinite number of red and blue sticks?

Gueda
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    Once you try to formalize "random" and "probability" you realize that there are some very subtle problems when you try to apply that to the natural numbers (or any countably infinite set). – Asaf Karagila Oct 04 '18 at 09:54
  • What is the pattern you are trying to establish in the sequence ${1, 4, 7, 10, ...}$? Are you listing every third integer starting with 1? There is certainly an argument about density within the naturals that could be done (if the pattern you give could be identified). One could define a density metric $\lim_{n\rightarrow\infty} \frac{N(n)}{n}$ where $N(n)$ is the number of elements in your set that are less than or equal to $n$. For example perfect squares have density zero in the naturals. – Michael Oct 04 '18 at 12:03
  • You recieved 2 answers to your question. Is any of them what you needed? If so, you should upvote all the useful answers and accept the answer that is most useful to you. – 5xum Oct 23 '18 at 07:47

2 Answers2

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You have to specify a distribution first ("what's the probability of picking stick $n$?"), and once you do that, it's a "simple" problem of summation.

Note that there is no uniform distribution, so you can't say that the probability is the same for each stick.

Arthur
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The answer depends on what the probability of picking sticks $1,2,3,4,5,6,\dots$ are. In order to answer the question, you have to first specify the answer to this question:

What is the probability of picking stick number $k$ in your case?

Note that you cannot, in your case, simply say "I pick each stick with the same probability", because, if the probability of picking each stick is $p>0$, then the probablitiy of picking one of the first $n$ sticks is $n\cdot p$, and for large enough values of $n$, this number will be greater than $1$, which is impossible.

This is what Arthur means in his answer when he says there is no uniform distribution, and this fact means that your question is actually much more complicated than it appears.

TL;DR: infinity is weird, and "picking a random item" from an infinite set is not a well defined action.

5xum
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