Die Roll Probability

Suppose that $N$ people (with $2\leq N\leq 15$ ) each roll a 20-sided die. The goal of this activity is to determine the probability that at least two people obtain the same die roll via simulations.

For a fixed $N$ , we will run the following procedure 10,000 times:

Select a random integer from 1 to 20, $N$ times.
Determine if at least one integer appears in the random sample more than once.
Return a 1 if the previous step is true and 0 otherwise.

We can then average our result to get an approximate value for the desired probability.

In [1]:

import numpy as np

In [23]:

def dieRoll(N):
    rolls = np.random.randint(1,21,size=N)
    for i in range(N-1):
        if rolls[i] in rolls[i+1:]:
            return 1
    return 0

In [33]:

def sim(N):
    print("The probability that at least two people out of " + str(N) + " roll the same number on a 20-sided die is approximately: " + str(sum([dieRoll(N) for i in range(10000)])/10000))

In [34]:

for i in range(2,16):
    sim(i)

The probability that at least two people out of 2 roll the same number on a 20-sided die is approximately: 0.0512
The probability that at least two people out of 3 roll the same number on a 20-sided die is approximately: 0.1465
The probability that at least two people out of 4 roll the same number on a 20-sided die is approximately: 0.2687
The probability that at least two people out of 5 roll the same number on a 20-sided die is approximately: 0.4192
The probability that at least two people out of 6 roll the same number on a 20-sided die is approximately: 0.5594
The probability that at least two people out of 7 roll the same number on a 20-sided die is approximately: 0.6943
The probability that at least two people out of 8 roll the same number on a 20-sided die is approximately: 0.804
The probability that at least two people out of 9 roll the same number on a 20-sided die is approximately: 0.8787
The probability that at least two people out of 10 roll the same number on a 20-sided die is approximately: 0.9367
The probability that at least two people out of 11 roll the same number on a 20-sided die is approximately: 0.9692
The probability that at least two people out of 12 roll the same number on a 20-sided die is approximately: 0.9848
The probability that at least two people out of 13 roll the same number on a 20-sided die is approximately: 0.9946
The probability that at least two people out of 14 roll the same number on a 20-sided die is approximately: 0.9984
The probability that at least two people out of 15 roll the same number on a 20-sided die is approximately: 0.9989