April 15, 2025

Natalie Groysman 

11th Grade

Brooklyn Technical High School

The birthday paradox, also known as THE birthday problem, is one of the most well-known examples of probability theory. It refers to the surprising result that in a group of just 23 people, there is a greater than 50% chance that at least two of them share the same birthday. Our natural intuition leads us to believe that the odds of two people in a relatively small group sharing the same birthday should be quite low. However, the reality is quite different, and the math behind the paradox reveals the surprising nature of probabilities in this scenario.

At first glance, the birthday paradox seems illogical. After all, there are 365 days in a typical year.  This causes us to assume that a large group of people would be necessary to ensure that at least two share the same birthday. This is where the "paradox" occurs, not in the result itself, but rather in the way our intuitive understanding of probability often fails us when dealing with real-world situations.

To understand the birthday paradox, one has to understand mathematical combinations and probabilities. When we consider whether two people share the same birthday, it’s a simple problem. However, as more people are added to the group, the complexity of the problem increases. With each new person, there are more possible pairings of birthdays, and the chances of a match rise much faster than expected. Continuing to increase n by 1 will increase the possibilities, and at 23 people, it will reach above 50% (equation below). 1*(1-1/365)*(1-2/365)*...(1-(n/365)).

The birthday paradox is not just an interesting theoretical idea; it has practical applications in a variety of fields, particularly in cryptography and computer science. One important application is in the concept of hash collisions. In cryptography, a hash function is used to convert data into a fixed-size string of characters, which is often used for security purposes such as password storage. The birthday paradox can be used to estimate the likelihood of two different inputs producing the same hash value—a situation known as a hash collision. Understanding the birthday paradox helps cryptographers design more secure systems by minimizing the risk of collisions.

The birthday paradox serves as a powerful example of how probability can defy our expectations. While it may seem improbable that two people in a small group would share a birthday, the math tells a different story. The fact that in a group of just 23 people, the probability of a shared birthday is greater than 50% often surprises those encountering the paradox for the first time. The birthday paradox highlights the importance of understanding probability in both theoretical and practical contexts, and it offers valuable insights into how we can better approach problems involving chance and uncertainty.