Python Code for Computing Generalized Birthday Problem

Problem Description

Generalized Birthday Problem: Consider $n$ people in a room. Assume each person’s birthday is equally likely to be any of the 365 days of the year (ignoring February 29), and all birthdays are independent. When $n$ and $r$ satisfy certain conditions, the probability that at least two people have birthdays within $r$ days of each other is:

Implementation

This is a simple code using Log-Probability Computation to calculate $P$. The logarithmic transformation helps prevent numerical overflow and reduces computation errors when dealing with large factorials.

import math

ln = math.log


def calculateP(n, r):
    up = 0
    dl = (n - 1) * ln(365)
    dr = 0

    for i in range(1, 365 - n * r):
        up += ln(i)

    for j in range(1, 365 + 1 - r * n - n):
        dr += ln(j)

    right = math.exp(up - dl - dr)

    return 1 - right


print(calculateP(10, 0))
print(calculateP(20, 0))
print(calculateP(23, 0))
print(calculateP(30, 0))
print(calculateP(40, 0))
print(calculateP(50, 0))

print()

print(calculateP(10, 1))
print(calculateP(20, 1))
print(calculateP(23, 1))
print(calculateP(30, 1))
print(calculateP(40, 1))
print(calculateP(50, 1))

print()

print(calculateP(10, 2))
print(calculateP(20, 2))
print(calculateP(23, 2))
print(calculateP(30, 2))
print(calculateP(40, 2))
print(calculateP(50, 2))
Code language: Python (python)

Results

Source

This code is used to solve a problem in the homework of Probability and Statistics for Information Science course. You can find the complete assignment in my GitHub repository (The visibility will be changed to public after the course ends).