In general your reasoning fails in this assumption:
However, since there are an infinite number of numbers in that interval, shouldn't the sum of all the probabilities sum up to infinity?
It's a mathematical problem, known since the Zeno of Elea Paradoxes.
Two of his claims were that
- An arrow can never reach its target
- Achilles will never overtake a turtle
Both of them were based on the claim that you can build an infinite sequence of positive numbers (in the former case by saying that an arrow has to fly infinitely times half of the remaining way to the target, in the latter by saying that Achilles has to reach position where the turtle was previously, and in the meantime the turtle moves to a new position that becomes our next reference base point).
Fast forward, this led to a discovery of infinite sums.
So in general sum of infinite many positive numbers does not necessarily have to be infinite; however, it may not be infinite only if (an extreme oversimplification, sorry about that) almost all of the numbers in the sequence are very close to 0, regardless how close to zero you request them to be.
Infinity plays even more tricks. The order in which you add elements of the sequence is important too and might lead to a situation that reordering gives different results!
Explore a bit more about paradoxes of infinity. You might be astonished.