Calculating the Average Minimum Length of Random Numbers From 0 - 1 that Adds up to Greater Than 1
If I rewrite genSequence so that it takes numTrials as an argument, then I get the following timing in CPython:

Python 3.6.4 (default, Dec 21 2017, 20:33:21) 
&gt;&gt;&gt; from timeit import timeit
&gt;&gt;&gt; timeit(lambda:genSequence(10**8), number=1)
2.71825759
62.77562193598715


Based on this, it would take about 10 minutes to compute genSequence(10**9). Possibly you just didn't wait long enough.

This kind of loop-heavy numerical code generally runs much faster if you use PyPy, which has a "just-in-time" compiler. I get more than ten times speedup with PyPy:

[PyPy 5.10.0 with GCC 4.2.1 Compatible Apple LLVM 9.0] on darwin
&gt;&gt;&gt;&gt; from timeit import timeit
&gt;&gt;&gt;&gt; timeit(lambda:genSequence(10**8), number=1)
2.71816679
5.389536142349243


On PyPy you should be able to carry out \$10^9\$ trials in under a minute (on my computer it takes 51 seconds).

Some review points:


The number 1000000000 is hard to read — it could easily be confused with 100000000 or 10000000000. I would write 10**9 to make it clear.
There's no need for the variable numberOfAttempts; you could just add one to storenums on each loop.
The name storenums is a bit vague. This is the total length of the random sequences generated so far, so a name like total_length would be clearer.
Similarly, the name genSequence is vague. This calculates the mean length of a random sequence, so a name like mean_sequence_length would be clearer.
The meaning of the constant 1 is not altogether clear. I would give it a name like target_sum.
When a loop variable like x is not used, it's conventional to name it _.
range(0,numTrials) can be written range(numTrials).


Revised code:

import random

def mean_sequence_length(trials, target_sum=1.0):
    """Return mean length of random sequences adding up to at least
    target_sum (carrying out the given number of trials).

    """
    total_length = 0
    for _ in range(trials):
        current_sum = 0.0
        while current_sum &lt; target_sum:
            current_sum += random.random()
            total_length += 1
    return total_length / trials