The Birthday Paradox, also known as the Birthday Problem, is a famous probability puzzle that explores the counterintuitive concept of how many people are needed in a room to find a pair of individuals with the same birthday. Despite what one might expect, the number is surprisingly low.

 The paradox can be stated as follows: What is the minimum number of people needed in a room to have a greater than 50% chance that at least two of them share the same birthday?

 To understand the paradox, let's break it down step by step:

 1. The Probability of No Shared Birthdays:
 Consider a room with only two people. The probability that the second person has a different birthday from the first is 364/365 (assuming a non-leap year). Now, if a third person joins, the probability that their birthday is different from the first two is 363/365 (as there are already two fixed birthdays). This pattern continues, so the probability of no shared birthdays with n people in the room is given by:
 P(n) = (365/365) * (364/365) * (363/365) * ... * [(365 - n + 1)/365]

 2. The Probability of Shared Birthdays:
 To find the probability of at least two people sharing a birthday, we need to subtract the probability of no shared birthdays from 1. So, the probability of shared birthdays with n people in the room is given by:
 P'(n) = 1 - P(n)

 3. Calculating the Minimum Number of People:
 We need to find the smallest value of n for which P'(n) is greater than 0.5 (or 50%). This will give us the minimum number of people needed in the room to have a greater than 50% chance of shared birthdays.

 Using mathematical calculations, we find that the minimum number of people required is 23. With 23 people, the probability of at least two people sharing a birthday is approximately 50.73%.

 To understand this result, we can simulate the scenario by repeatedly generating random birthdays for a group of people. By running these simulations thousands of times, we consistently find that around 23 people are needed to achieve a greater than 50% chance of shared birthdays.

 The Birthday Paradox has been proven mathematically and is widely studied in probability theory. It was first introduced by Richard von Mises in 1939 and later popularized by the American mathematician Martin Gardner.

 References:
 1. Von Mises, R. (1939). Probability, Statistics and Truth. New York: Macmillan.
 2. Gardner, M. (1957). The fantastic combinations of John Conway's new solitaire game "life". Scientific American, 223(4), 120-123.