# The Birthday Paradox and the Increase in September Births

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.

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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.

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