What is the probability that, in a set of \(n\) randomly chosen people, at least two people share a birthday? The question is innocent, but the birthday paradox says that the probability exceeds \(50\%\) in a set of just 23 people. The result is counter-intuitive, as the product of probabilities is often counter-intuitive, but it’s important to remember that there are \((23 \times 22)/2 = 253\) pairs for a set of 23. Mathematically, in a set of \(n\) people, the probability that at least two people share a birthday is:

\[P(n) = 1 - \prod_{i = 0}^{n - 1} \frac{365 - i}{365},\]

for which we detail some specific probabilities:

\(n\) \(P\) (%)
1 0
5 2.7
10 11.7
15 25.3
20 41.1
23 50.7
70 99.9

Python Implementation

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import numpy as np
import matplotlib.pyplot as plt

def pTwoShareBday(num):
    
    p_ar = (365 - np.arange(num))/365
    prod = 1
    prod_ar = []
    for elem in p_ar:
        prod = prod * elem
        prod_ar = np.append(prod_ar, prod)
    print("The probability that at least 2 people share a birthday in a set\
of {:.0f} (randomly chosen) people is {:.2f}%.".format(num, 100*(1 - prod)))
    
    plt.figure()
    plt.plot(np.linspace(1, num, num), (1 - prod_ar), '.r')
    plt.xlabel('Number of People')
    plt.ylabel('Probability that at least 2 Share a B-Day')
    plt.show()
Birthday Paradox up to 23 People
Birthday Paradox up to 23 People
Birthday Paradox up to 100 People
Birthday Paradox up to 100 People

Mathematica Implementation

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pTwoShareBday = 1 - Product[((365 - i)/365), {i, 0, n - 1}];
ListPlot[Table[pTwoShareBday, {n, 1, 100}], 
 AxesLabel -> {"# of People in a Room", "P(2 Share a Birthday)"}]
BdayList = Table[N[Refine[pTwoShareBday, n == k]], {k, 1, 23}]