The Birthday Paradox
An Exercise in Probability Magic
A version of this article appears in ‘Twisted Logic: Puzzles, Paradoxes, and Big Questions’, by Leighton Vaughan Williams. Chapman & Hall/CRC Press. 2024.
SIZE MATTERS
What is the minimum number of individuals that need to be present in the room for it to be more likely than not that at least two of them share a birthday? This is what the ‘Birthday Paradox’ (or ‘Birthday Problem’) seeks to solve.
For the sake of simplicity, let’s assume that all calendar dates have an equal chance of being someone’s birthday and let’s disregard the Leap Year occurrence of 29 February.
A BASIC INTUITION: ANALYSING THE ODDS
At first glance, you might think that the odds of two people sharing a birthday are incredibly low. In a group of just two people, the likelihood of them sharing a birthday is a mere 1/365. Why is that? We have 365 days in a year, hence there’s only one chance in 365 that the second person would have been born on the same specific day as the first person.
Now, let’s take a group of 366 people. In this case, it’s certain that at least one person shares a birthday with someone else, due to the simple fact that we only have 365 possible birthdays (ignoring Leap Years).
The initial intuition may suggest that the tipping point—the group size at which there’s a 50% chance of two individuals sharing a birthday—is around the midpoint of these two extremes. You may think it lies around a group size of about 180. However, the reality is surprisingly different, and the actual answer is much smaller.
THE CALCULATIONS: UNRAVELLING THE BIRTHDAY PARADOX
To understand the concept better, we need to dig deeper into the probabilities involved. Let’s consider a duo: Julia and Julian. Let’s assume that Julia’s birthday falls on 1 May. The chance that Julian shares the same birthday, assuming an equal distribution of birthdays across the year, is 1/365.
What about the probability that Julian doesn’t share a birthday with Julia? It’s simply 1 minus 1/365, or 364/365. This number illustrates the chance that the second person in a random duo has a different birthday than the first person.
Adding a third person into the mix changes things slightly. The chance that all three birthdays are different is the chance that the first two are different (364/365) multiplied by the probability that the third birthday is unique (363/365). So, the probability of three different birthdays equals (364/365) × (363/365).
As we expand the group, the calculations continue in a similar manner. The more people in the room, the greater the chance of finding at least two people sharing a birthday.
Consider a group of four people. The probability that four people have different birthdays is (364 × 363 × 362)/(365 × 365 × 365). To find the probability that at least two of the four share a birthday, we subtract this number from 1. Thus, the odds of having at least two people with the same birthday in a group of four are about 1.6%.
As the number of people in the room increases, the probability of at least two sharing a birthday grows:
With 5 people, the probability is 2.7%.
With 10 people, the probability is 11.7%.
With 16 people, the probability is 28.1%.
With 23 people, the probability is 50.5%.
With 32 people, the probability is 75.4%.
With 40 people, the probability is 89.2%.
THE PARADOX UNVEILED: IT’S NOT JUST ABOUT BIRTHDAYS
You might be wondering why we need just 23 people to reach a 50% chance of finding shared birthdays. This can be explained by how many possible pairs can be made in a group. In a group of 23, there are 253 unique pairs. Each of these pairs has a 1/365 chance of sharing a birthday, and all these possibilities add up. This is what makes the birthday problem so counterintuitive. Basically, when a large group is analysed, there are so many potential pairings that it becomes statistically likely for coincidental matches to occur.
This is a perfect demonstration of the concept of multiple comparisons and an example of the so-called ‘Multiple Comparisons Fallacy’.
The same reasoning applies to balls being randomly dropped into open boxes. Assume there is an equal chance that a ball will drop into any of the individual boxes, and there are 365 such boxes, into which 23 balls are randomly dropped. There is an an equal chance, we assume, that a ball will drop into any specific box. Now, there is just over a 50% chance in this scenario that there will be at least two balls in at least one of the boxes. Randomness produces more aggregation than intuition leads us to expect.
YOUR PERSONAL BIRTHDAY CHANCES: WHERE DO YOU STAND?
The reason for the paradox is that the question is not asking about the chance that someone shares your particular birthday or any particular birthday. It is asking whether any two people share any birthday.
While the birthday problem shows the increased likelihood of shared birthdays in a group, the chance that someone shares your birthday specifically is a different question.
In a group of 23 people, including yourself, the probability that at least one person shares your birthday is much lower than 50%—it’s about 6%. This is because there are only 22 potential pairings that include you.
Even in a group of 366 people, the probability that someone shares your specific birthday is only around 63%.
CONCLUSION: THE MAGIC OF PROBABILITY AND THE BIRTHDAY PARADOX
The Birthday Paradox reveals an intriguing counterintuitive fact about probability: a group of just 23 people has a greater than 50% chance of including at least two people who share the same birthday. It sheds light on the intricacies of probability by demonstrating how many opportunities there are for matches to occur, even in seemingly small groups. For example, if you can find out the birthdays of the 22 players at the start of a football game, and the referee, more than half of the time two of them will share a birthday.
This fascinating concept has applications way beyond birthdays. It’s also very important for the safety and performance of computer systems and online security. This idea helps specialists prevent and deal with issues that occur when data unexpectedly overlaps. Understanding the paradox is crucial, therefore, for those who design and secure computer systems, helping them to make these systems more reliable and efficient.
Nevertheless, it’s in the social setting of parties where the paradox becomes a delightful surprise. Next time you’re among friends or at any casual meet-up, consider introducing this paradox; you might just bring to life the unexpected magic of probability!