I have two infinite sets of events $A$ and $B$ with the following probabilities:
Note that I have divided them into two sets only because it is easier to depict the probabilities; other than that, these two sets are essentially "indistinguishable".
I want to prove that the probability that any one of the events will take place is smaller than $1$.
I've figured I can use Inclusion-Exclusion principle, but the problem, of course, is that I have an infinite amount of events so I'm not quite sure how to accomplish that.
I tested it using the following Python script, so I believe that the probability of one of the events taking place is indeed smaller than $1$:
from decimal import Decimal from decimal import getcontext getcontext().prec = 100 def getBits(num): bit = 0 bits =  while num > 0: if num & 1: bits.append(bit) bit += 1 num >>= 1 return bits def prod(arr): res = Decimal(1) for val in arr: res *= val return res SIZE = 20 sums = [Decimal(0) for k in range(SIZE)] A = [Decimal(2)/(6*k-1) for k in range(1,SIZE)] B = [Decimal(2)/(6*k+1) for k in range(1,SIZE)] probabilities = [p for pair in zip(A,B) for p in pair] for n in range(1,1<<SIZE): bits = getBits(n) sums[len(bits)-1] += prod([probabilities[bit] for bit in bits]) print(sum([sums[k]*(-1)**k for k in range(SIZE)]))
After a minute or two, the printout shows a probability of approximately $89\%$.
Does anyone see a way for me to achieve my purpose mathematically?
Again - my primary objective is to prove that the probability of one of the events taking place is smaller than $1$. I don't mind calculating the exact value of this probability along the way, but if there's another way which only proves that it is smaller than $1$ then I'm fine with that.
Thank you very much!