Here's the cowboy method of brute simulation (probably my favorite way of exploring problems). I don't think I did this on the extra credit because of my easier shower idea (plus it was EC and the homeworks were kind of large).

Anyways, I was super curious to see if empirically the answer seemed to be distributed around a 1.75 value, or the 1.565 value.

First, generating n random values:



In [3]:

    
from random import randint

def gen_rand_nums(n):
    for i in range(n):
        yield randint(1,100)

Then to run trials (trying to keep this simple and dumb to catch bugs---I find generating random things often tricky):



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def rand_trial(n):
    if n <= 50:
        return 1
    if n <= 75:
        return 2
    if n <= 100:
        return 3

Now for running the whole thing:



In [6]:

    
def run_trials(n):
    product = 1
    for i in gen_rand_nums(n):
        product *= rand_trial(i)
    return product ** (1 / n)

And some test runs/output:



In [12]:

    
print(run_trials(100))
print(run_trials(250))
print(run_trials(500))
print(run_trials(1000))
print(run_trials(1200))
print(run_trials(1300))









    



1.4892039780975725
1.5584266462422762
1.591381491721272
1.563631740023799
1.5638235281980817
1.5712215952419577



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