If it's a random real then the probability is zero.
If it's an arbitrarily close approximation to a random real then the probability is arbitrarily close to zero.
How can the sum of infinitely many zero probabilities be 1? I can understand how the sum of infinitely many infinitely close to zero values can be 1, but not infinitely many exactly zero values
There's no such thing as a "sum of infinitely many" anything. What we are talking about is the limit of an infinite series, which behaves nothing at all like a sum.