Imagine yourself in a room, surrounded by 22 other people. You might have nothing in common with these 22 people, not a single piece of similar background. What might surprise you is that there is a 50 percent chance that at least two people in the room share a birthday. Up the room to 50 people, and that same probability becomes 97 percent.
Of course, this doesn’t seem like it would be true. Take any classroom of students or meeting at an office: there’s no way that half of them have at least two people sharing birthdays… right? After all, there are 365 possible birthdays excluding the leap day, so any overlap between two people should be rare. In fact, the probability of two people having the same birthday is 1/365, or under 0.3%. The issue with this logic arises when we add a third person, who now has to avoid 2 different birthdays instead of just one. Similarly, adding a fourth person would require avoiding 3 other birthdays. As we keep on adding members to our group, the amount dates taken up by birthdays increases and restricts the remaining possibilities.
To further grasp this concept, we can look at the math behind the birthday paradox. Looking at just one person, there is clearly no possibility of matching birthdays, as there is no one to match with. We can successfully add person two with a probability of 364/365, and then person 3 with 363/365, and so on. In fact, we can add person n with a probability of (366-n)/365, as they must avoid the birthdays of the previous n-1 people. When we multiply this together, we get that with n different birthdays, the probability of overlap is (365 P n)/(365^n). Sure enough, plugging in n = 23 results in about a 49.3% percent chance that no two people share a birthday. The rate of no overlap decreases at a faster pace every time a new person is added and the chance of shared birthdays grows exponentially.
The birthday paradox serves as a warning against excessive risk-taking behavior. While you might leave unscathed after one potentially dangerous action, the riskier you get the more assured your downfall becomes.