E Burch
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.
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.
I hope that 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!