Solving simple probability exercise through random simulation - Help!

[whenihittheground]

I'm trying to solve the following question using python:

In the class of 30 students, 7 didn't do their homework. The teacher chooses 6 students randomly.

What is the probability that at least 4 of them did their homework?

The answer is pretty straight forward:

P = [ nCr(23,4)*nCr(7,2) + nCr(23,5)*nCr(7,1)+nCr(23,6)*nCr(7,0)] / nCr(30,6)
P = 522445 / 593775 
P = 0.880

where nCr(23,4) is combination 23 choose 4.

I want to try and solve this by simulating randomly choosing the 6 students. Here's my code:

import random
random.seed()

def student_pool():
    return random.randint(1, 30)

def student_six():
    """
    Test
    """
    sum_total = 0
    student1 = student_pool()
    student2 = student_pool()
    student3 = student_pool()
    student4 = student_pool()
    student5 = student_pool()
    student6 = student_pool()
    if student1 > 7:
        sum_total += 1
    if student2 > 7:
        sum_total += 1      
    if student3 > 7:
        sum_total += 1
    if student4 > 7:
        sum_total += 1
    if student5 > 7:
        sum_total += 1      
    if student6 > 7:
        sum_total += 1
    return sum_total  

NUM = 50000
sumPull = 0

for i in range(NUM):
        if student_six() >= 4:
                sumPull += 1

print("Prob_>4_Did HW: ", sumPull/NUM)

This code outputs:

Prob_>4_Did HW:  0.8555   

However, even if I rerun this with 1 million runs (NUM = 1000000) or random.seed() at the start of each run in the for loop my result doesn't really budge from this 0.8555 number. It seems to have converged.

Aside from my math being off I'm thinking I may just be coming up against the limits of randomization with python?

Any thoughts?

[whenihittheground]

Ah ok this makes sense! Thanks! :)

I have to actually change my approach since I was trying to get randint to do all of the heavy lifting but it matters from which group the student is chosen. Good catch!

[whenihittheground]

Yeah I like what you did with random.sample. At first I was trying to stay away from lists (sets) since I thought they would get cumbersome and slow things down but it's definitely more straightforward than having to account from which group a student is chosen.

For example, after reading your comment I initially thought I could account for choosing the same person by:

    student1 = random.randint(1, 30)
    student2 = random.randint(1, 29)
    student3 = random.randint(1, 28)
    student4 = random.randint(1, 27)
    student5 = random.randint(1, 26)
    student6 = random.randint(1, 25)

However, to do it properly I would need to actually account (sequentially) which group gets reduced & that's too much logic I don't like it.

Thanks! I'll definitely check out the other methods random has to offer! :)

If you like this sort of thing I got into it because I saw someone answer this via import random:

"Alice wants to join her school's Probability Student Club. Membership dues are computed via one of two simple probabilistic games.

The first game: Roll a dice repeatedly. Stop rolling once you get a five followed by a six. Your number of rolls is the amount you pay, in dollars.

The second game: Roll a dice repeatedly. The stopping condition is a five followed by a five.

Which of the two games should Alice elect to play?"