Buzz
The odds of two people sharing a birthday in a group of five friends are quite low. How many people would you need to have a 50% chance of this occurring?How many people share your birthday? For years, I didn’t know anyone with the same birthday as me, but as my circle grew, the likelihood of finding someone with the same birthday increased. Now I know at least five others who share my summer birthday. What are the odds?
How Does the Birthday Paradox Work?
The answer lies in the birthday paradox: How many people must be in a random group for there to be a 50% chance that at least two individuals share a birthday?
Consider a classroom of 30 children, each with a birthday randomly assigned from 365 days in a year. The probability that two children share the same birthday seems quite low, right? After all, with 30 children randomly distributed across 365 days, it seems unlikely that any two would have the same birth date.
So, how many people does it take for two to share a birthday? Many people might quickly assume that 182 is the correct number, roughly half the days in a year. But is it really necessary to have 182 people for two of them to have the same birthday?
Nope, it's not that simple: The birthday paradox involves exponential probabilities.
The Probabilities of the Birthday Paradox Are Exponential
"The key point is that people tend to vastly underestimate how quickly the probability grows as the group size increases. The number of potential pairings grows exponentially with each additional person in the group. And humans struggle to grasp exponential growth," said Jim Frost, a statistician and columnist for the American Society of Quality's Statistics Digest, in an interview with Live Science.
We're just not very good at predicting probabilities, especially when they are as surprising as the birthday paradox.
"I find these types of problems fascinating because they show how people often misunderstand probabilities, leading to poor decisions or conclusions," said Frost.
To determine how many people are needed for two to share a birthday, we must perform the calculations and start eliminating possibilities.
For a group of two people, the odds that one shares a birthday with the other are 364 out of 365 days, or about 0.27%. Add a third person, and the chances increase to 363 out of 365 days, or approximately 0.82%.
The Answer to the Birthday Paradox
As you might have predicted — and correctly, I might add — the bigger the group, the higher the probability that two people share the same birthday. So, what exactly is the answer to the birthday paradox? By crunching the numbers, we find that with a group of 23 people, there is about a 50 percent chance that at least two individuals will have the same birthday.
Why does 23 seem so unexpectedly low? The key is in the nature of exponents. Our minds don't typically grasp the exponential effect when doing mental calculations. We often assume that probability calculations are straightforward and linear, which is a misconception.
In a room with 22 other people, if you were to compare your birthday to the birthdays of the others, you would only have 22 comparisons to make.
However, if all 23 people’s birthdays are compared against each other, there are many more possible comparisons. How many exactly? The first person makes 22 comparisons, the second person compares with the first and then only 21 others, the third person compares with 20, the fourth with 19, and so on. Adding them all together gives a total of 253 possible birthday comparisons or combinations. Thus, with 23 people, there are 253 opportunities for two birthdays to coincide.
This graph illustrates the chance of having at least one matching birthday between two people in a group. The probability rises with the number of people involved.
Source: Wikimedia Commons (CC BY SA 3.0)Here’s a similar exponential growth problem to the birthday paradox: If you're offered 1 cent on the first day, 2 cents on the second, 4 on the third, 8 on the fourth, and so on for 30 days, it seems like a bad deal. However, due to exponential growth, by the 30th day, you’d accumulate $10.7 million.
These types of probability problems highlight the value of mathematics in enhancing our understanding of the world. While their outcomes may seem counterintuitive, they serve to demonstrate how mathematics can improve our lives.
The next time you're in a group of 23 people, rest assured there’s a 50% chance that two of you will share the same birthday.
From a psychological standpoint, the brain uses two distinct "systems" for problem-solving and decision-making: The first system is intuitive, enabling quick decisions, while the second system involves thoughtful and sometimes extended reasoning to find an answer. The birthday paradox depends on the second system to calculate and arrive at the correct conclusion.
