The Birthday Paradox: More Than One Coincidence?

We’ve all experienced the delightful surprise of meeting someone who shares our birthday. It’s a coincidence that often sparks conversations and connections. But have you ever paused to think about the intriguing mathematics behind this seemingly everyday occurrence?

Let’s start with a simple scenario. If my birthday falls on a specific day, the probability that I meet someone with the same birthday is, at first glance, 1 in 365. Easy, right? But what if we introduce a third person into the mix? Determining the likelihood that any two of us share a birthday becomes a bit more complex.

Here’s where it gets interesting. To calculate the probability that in a group of three people, at least two share the same birthday, we can’t just add another 1 in 365 chance. Instead, we have to consider the probability that none of the three share a birthday and subtract that from 1. Confused? Let’s break it down.

First, we consider the probability of the second person not sharing a birthday with the first, which is 364/365. Then, the probability that the third person doesn’t share a birthday with the first two is 363/365. Multiplying these probabilities together and taking the product away from 1 (since this gives the probability of not sharing a birthday) still doesn’t give a very interesting result. There’s roughly a 0.8% chance that any two people in this group of three share the same birthday.

However, as we expand the group, the odds of shared birthdays skyrocket. By the time we reach 23 people, the likelihood that two of them share a birthday exceeds 50%. What makes this phenomenon so unintuitive is that we're not merely comparing the birthdays of 23 people against one individual’s birthday. Instead, it’s a complex web of comparisons among all birthdays within the group. When there are 50 people, the probability soars to about 97%, and with 75 people, it reaches 99.97%.

We can simplify this to a general formula where P(N) is the probability of there being two birthdays in a group of N people:

So, why does this matter? These principles have a fascinating application in computer science, particularly in internet security. When you create a password, it is converted to something called a ‘hash’, which is essentially a long string of letters and numbers which represents the original password. This prevents your passwords from being read by a human, keeping them secure. One specific hashing algorithm, the MD5 message-digest algorithm, was vulnerable to attacks rooted in the same principles as the birthday paradox.

Hash collisions occur when two distinct inputs produce the same hash output; hackers exploited this vulnerability by leveraging the mathematics of the birthday paradox. By efficiently finding hash collisions, they could decipher the encoding algorithm, compromising internet security.

Fortunately, understanding the complexities of the birthday paradox ultimately contributed to strengthening internet security protocols. As computer scientists recognized the vulnerabilities, they devised more robust encryption methods, ensuring our digital lives remain protected against potential threats.

The birthday paradox is a reminder that mathematics can serve as a framework through which we can prove extraordinary results.