e from random sums

select numbers between 0 and 1 randomly until sum is > 1

The expected # of selections needed is equal to e

# setup imports
import math
from typing import Tuple

import numpy as np
import pandas as pd
from tqdm.notebook import trange

def run_experiment() -> Tuple[float, int]:
    s = 0
    n = 0

    while s <= 1:
        n += 1
        s += np.random.uniform(0, 1)

    return s, n


logs = [run_experiment() for _ in trange(2_000_000)]

df = pd.DataFrame(logs, columns=["sum", "nums_to_one"])
df.head()

	sum	nums_to_one
0	1.160454	2
1	1.099141	2
2	1.171709	3
3	1.664728	3
4	1.291741	2

euler's number: 2.718282
avg num to one: 2.717664
     avg sum*2: 2.718284

df.describe()

	sum	nums_to_one
count	2.000000e+06	2.000000e+06
mean	1.359142e+00	2.717664e+00
std	2.425132e-01	8.745580e-01
min	1.000000e+00	2.000000e+00
25%	1.152672e+00	2.000000e+00
50%	1.325424e+00	2.000000e+00
75%	1.538078e+00	3.000000e+00
max	1.999527e+00	9.000000e+00