Say you're in a math class, and there are 23 students in the class. One day, the professor states that the odds are likely that two students in the class share the same birthday.
With 365 possible birthdays if you eliminate February 29th, and only 23 students, that can't be correct, but it is. Welcome to the Birthday Paradox.
How the paradox works
In the field of probability, the sum of all the possible outcomes, which is called the sample space, is always equal to 1, or 100%.
We also know that there are two possible outcomes to the Birthday Paradox:
Outcome #1 - At least two people share a birthday, or
Outcome #2 - No two people share a birthday.
Therefore, Outcome #1 = 100% - Outcome #2.
Now, let's work out the chances of Outcome #2, that no two people share a birthday. The first student, Student A, can have any birthday, so his or her probability is 365/365. For no two students to share a birthday, the second student, student B, has 364/365 possible birthdays, and the third student, Student C, only has 363/365 possible days, all the way down to Student W, who has 343/365.
If we multiply all these terms together, we get 0.4927, or a 49.27% chance that no two students share a birthday. This is Outcome #2 that we defined above. 100% - 49.27% = 50.73%, which is Outcome #1, that two students share a birthday. Those odds are better than 50-50, and the professor was indeed correct.
This surprising result is because of combinatorics, a field of mathematics concerned with counting. For example, a group of 5 people has 10 possible pairs, while a group of 10 people has 45 possible pairs. A group of 23 people has 253 possible pairs, which is over half the number of days in a year. In a group of 70 people, there are 2,415 possible pairs, and the probability that two people share a birthday is a whopping 99.9%, or a virtual certainty.
The number of possible pairs grows quadratically, that is, it is proportional to the square of the number of people in the group.
Actual birth date distribution
The heat map below shows the actual distribution of births in the U.S. between 1994 and 2014 as collected by the U.S. Social Security Administration.
The map shows a real spike in births during the month of September, with the number one and number two days being September 9th and September 19th. Given that human gestation takes around 280 days, this data provides a whole new insight into how people really celebrate the December holidays.
The Outliers effect
Since 2008, when Malcolm Gladwell published his wildly popular book Outliers, there has been a spike in September births. In the book, Gladwell made the case that children who are older in each grade are better developed mentally, emotionally and physically, and thus do better.
In most U.S. states and Washington D.C., the cut-off birthdate for incoming classes is September 1st. That means that children born in September will be the oldest in their class.