If there are 23 randomly chosen people in a room, what is the probability that two of them have their birthday on the same day? The answer, surprisingly, is 50.7%. What’s more, when you increase the number of people to 50, the probability becomes 97%! It may seem ridiculous, but it is actually true. The key to understanding why this is the case lies in calculating the probability.
The scenario
Let’s number our 23 people with numbers 1-23 and let them into our room one by one.
Each time a person enters the room, we calculate the probability that the person entering the room does not share a birthday with the people already present in the room. When the first person enters, the room is empty so therefore we can say that it is 100% improbable that they could share a birthday with the 0 people in the room.
When the second person enters the room, we can say that there is a ³⁶⁴⁄₃₆₅ likelihood that they have different birthdays according to the days in the year. Similarly, when the third person enters the room, the probability becomes ³⁶³⁄₃₆₅. For the third person, the probability is ³⁶²⁄₃₆₅, becoming slowly more and more likely as more people enter the room.
Therefore, we can multiply all these different individual probabilities together to calculate the total probability that no people in our fictional room share the same birthday:
Probability = 1 x ³⁶⁴⁄₃₆₅ x ³⁶³⁄₃₆₅ x ³⁶²⁄₃₆₅ …
Note that the ‘1’ at the beginning represents our initial 100% probability.
If you follow this formula to work out the improbably, you can then subtract your results from 1. These are probabilities that two people share a birthday for a certain group of people:
- 10 People – 12%
- 20 People – 41%
- 23 people – 50.7%
- 30 people – 70%
- 50 people – 97%
- 100 people – 99.99996%
- 200 people – 99.99999999999999999999999999998%
- 300 people – 1 – (7×10⁻⁷³)
- 350 people – 1 – (3×10⁻¹⁷¹)
- ≥366 people – 100%
As you can see, there’s a 70% chance that in a class of 30 people, two people share the same birthday.
This classic probability problem completely breaks our human intuition about odds and luck. If you are in a room with a small group of people, what are the chances you would expect two of them share the exact same birthday (same day and month)? Most people think you’d need at least 100 or 200 people to get a decent chance since there are 365 days in a year. However, as shown, the maths says you only need 23 people in a room for the odds to hit a 50.7% chance. And if you get 57 people together, the odds skyrocket to 99%. Here is why our brains get this so wrong.
The trap of thinking about yourself
When you hear this problem, your brain automatically asks ‘What are the chances that someone in this room shares a birthday with me?’ That is a completely different maths problem. If you want to find someone who shares your specific birthday, you do need a massive crowd. But the paradox isn’t asking about you. It’s asking if any two people in the room match. It does not matter if they are both born on March 12th, August 30th, or Christmas Day. Any match wins.
The power of pairs
Instead of counting people, the we counts pairs. If you are in a room with 22 other people, you do not just compare your birthday to theirs (22 comparisons). Everyone compares their birthday with everyone else. To find out how many unique pairs can be made in a room of 23 people, we use a basic combination formula that multiplies 23 by 22, and then divides by 2. This gives us exactly 253. In a room of just 23 people, there are 253 different pairs of people who could potentially share a birthday. Since 253 is more than half of the total days in a year, it starts to make sense why the odds of a match hit over 50%.
Flipping the maths upside down
To actually calculate the exact probability, mathematicians do not try to add up all the ways people can match. That gets messy. Instead, they calculate the odds that everyone in the room has a completely unique birthday, and then subtract that from 100%. The maths works by looking at the shrinking options for each new person added to the room. The first person can have any birthday. The second person has a slightly smaller chance of landing on a unique day, the third person has an even smaller chance, and so on. When you chain these diminishing chances together for all 23 people, you find there is about a 49.3% chance that everyone in the room has a completely unique birthday. Since the total probability must equal 100%, you subtract that 49.3% from 100%. This leaves a 50.7% probability that at least two people in the room share a day. It feels like magic, but it’s just the hidden power of combinations.
Hopefully this has helped you understand a little more about the Birthday Paradox, and the next time you are at school, at work, at a party or any other gathering, make sure to test this theory in the real world and amaze everyone!