How do you add the results together each time a function is called?

[ichabod801]

First, a list would be a better way to store the results. If you have counts = [0] * 11, then you can avoid the if/elif chain. I would also eliminate looping through twice:

for num in range(1000):
    counts[random.randint(1, 10)] += 1

Then a simple list comprehension gets the percentages:

percentages = [count / 1000 for count in counts]

Note that your ability to use list_length in a function is not inexplicable. If Python can't find the value in the function scope, it will check the module/global scope for it. That is, as long as you haven't done an assignment to that name, which flags the name as local scope to Python.

Then you can check that sum(percentages) == 1. I expect it will be wrong due to the minor inaccuracies of floating point math, but it should be close.