Sometimes, an event is more likely to occur than we think. For example, if you survey a random group of 23 people, you have a 50–50 chance of finding someone with the same birthday as you. Understanding the Birthday Paradox is all about mathematical probability.
What’s the key to understanding the Birthday Paradox?
Understanding the Birthday Paradox is important to understanding the limits of our intuition.
The Birthday Paradox is a statistical phenomenon that states that in a group of just 23 people, there’s a 50-50 chance that two of them will have the same birthday. This likely seems counterintuitive. The probability of any two people having the same birthday seems much lower, right?
But as the number of people in a group increases, the probability of two people having the same birthday also increases. (That part makes sense.)
To understand the Birthday Paradox, we have to consider the probability of two people not having the same birthday.
If the first person in a group has a birthday on any day of the year, the probability that the second person does not have the same birthday is 364/365, or 0.9973. The probability that the third person does not have the same birthday as the first two people is 363/365, and so on.
As the number of people in the group increases, the probability of any two people not having the same birthday decreases.
How does the math work on this?
Ok, let’s say we have a group of 23 people. The probability that any two people do not have the same birthday is (364/365)^(23*22/2) = 0.4927. That means that there is a 50.73% chance that two people in the group will have the same birthday.
In a group of 30 people, the probability increases to 0.7037. In that case, there is a 29.63% chance that two people will have the same birthday.
A simpler way to think about the Birthday Paradox is to think about it as a game of matching pairs. If you have a deck of cards with 365 cards and you randomly draw 23 cards, the probability of matching pairs is 50.73%. The more cards you draw, the higher the chance of matching pairs. — WTF fun facts
Source: “Probability and the Birthday Paradox” — Scientific American