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?
On
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.
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.