Chances That Your Coworker Is Your Birthday Twin

Does Your Birthday Match Your Friends?

My name is Michael, and I wanted to solve the birthday problem.
August 30. Although this is my birthday, to most, this date is just a random calendar date. However, to a select few, it is a match to your birthday. Let's proceed.

Now, if you walked up to someone and asked when their birthday is, the chances that it will match yours is extremely low. What if you walked up to 2 people? Now the chances that 2 out of the 3 of you have the same birthday is somewhat higher, but still very low.

So, how many people do you need in a room to ensure that there is greater than a 50% chance that 2 of those people share a birthday? Hint: The answer is fewer than you would expect.

Yes, there is a 1 in 365 chance that you and I have the same birthday. This is a result of the following: the probability of being born on any one day of the year is 1 or 365/365 (considering it is not a leap year). Now, the chances of walking up to a person and having the same birthday is 1/365. This is because Person A must have the same birthday as Person B. There is 1 possibility that this happens out of 365 dates. So, this probability equates to 365/365 x 1/365 = 1 x 1/365 = 1/365 = 0.0027 = 0.27%, which is a very small chance you will share a birthday with a random stranger. Another way to calculate this is to find 1 minus(-) the probability that no one shares a birthday. 1 - 365/365 x 364/365 = 0.27%. This method will make it significantly easier as the number of people you ask increases.

Now, what if you tested your skills and walked up to not 1, but 2 people and asked for their birthdays? Now the probability is slightly higher. It can be calculated by figuring out the chances that no one shares a birthday. This method of working backwards helps significantly since that are 365 days in the year, and we do not know which 2 of the 3 will have the same birthday. So, the first person can be born on any of the 365 days. Since the second person will not have the same birthday, there options are out of 364, and so on. So, the probability that all 3 people have different birthdays is 365/365 x 364/365 x 363/365 = 0.9918 = 99.18%. But since we are looking for chances of the same birthday, we subtract the value from 1. 1 - 0.9918 = 0.00820 = 0.82%.

So, the addition of people raises the probability that 2 of those people share a birthday.

You may be wondering who the mastermind behind this concoction was. Well, that is open to interpretation. However, it is widely recognized that Richard von Mises first devised the "Birthday Problem" around 1920 just after graduating from Harvard University. From the "Selected Paper of Richard von Mises," the idea to this birthday philosophy came when Mises once noticed that 3 people out of his 60 person class shared a birthday. He then used this principle to devise the expected number of birthday repetitions as a function based on number of people.

The main system used by Mises to find a way to see how often birthday repetitions are apparent was to see how often birthdays don't repeat. By starting at 365 days for the first person's birthday, subtracting 1 from each additional person allowed Mises to uncover percentages of birthday reoccurrences.

When finding the probability of a repeated birthday, it becomes clear that 23 people are needed to ensure a greater than 50% chance that 2 people share a birthday. Using the formula as proof,

.

So, this finding by Mises is accurate.



This chart depicts the probability of a repeating birthday based of n, the number of people, to give p(n), the percentage of a double birthday in the group.

Although the odds of a matching birthday start off low, once a group of 20-40 people are present, the chances spike up high. Once at 40 people, there is roughly a 90% chance of a matching pair. Of course, since 100% is the highest chance there is a pair of birthdays within a group, the probability flatlines around 70 people. After 100 people, there is roughly a 99.99997% chance that there is a pair. This means there is a 0.00003% chance there is not a pair. To put this into perspective, National Weather Service states that the chances of getting struck by lightning in your lifetime is 1 in 14,600 or 0.000068%. This justifies that you are twice as likely to get struck by lightning than to not have a birthday twin in a group of 100. Although, some would originally think you would need almost 350 people to be that sure since there are 365 days in a year, you actually only need 100. The odds are in favor of the group of 100. I'd place my money with them.


The graph shows the spike of a birthday match from a range of 20-40 people, and the flatline around 55 people in a room.

Based on my interest in probability and this specific math principle, I decided to conduct my own study. I wanted to see how likely the 50.73% repeated birthday chance appears in the real world. I recently reached out to my peers in high school, and asked them to fill out a form that asked for their birthday. Out of the 186 responses, I was able to create a method of confirming the 23 person principle. After assigning each student response a number 1-186 based on the order of submission, I used a random number generator that will provide 23 numbers out of 186. Then, I wrote out the birthdays of the 23 people chosen and record whether a repeated birthday occurred.

This experiment was repeated 50 times for precision. Based on the 50.73% chance that there is a repeated birthday out of a group of 23, it is expected for a repeated birthday to occur 25.365 times out of 50. Since a fraction of a repeated birthday occurrence is not possible, I hypothesize that a repeated birthday will happen around 24-27 times.

The results are as follows. If a repeated birthday occurred out of the 23 people randomly chosen, the result was Yes, and if no repeated birthday happened, the result was No.

Test 1	Test 2	Test 3	Test 4	Test 5	Test 6	Test 7	Test 8	Test 9	Test 10
Yes	Yes	No	No	Yes	Yes	Yes	Yes	No	Yes

Test 11	Test 12	Test 13	Test 14	Test 15	Test 16	Test 17	Test 18	Test 19	Test 20
Yes	No	No	Yes	No	No	Yes	Yes	Yes	Yes

Test 21	Test 22	Test 23	Test 24	Test 25	Test 26	Test 27	Test 28	Test 29	Test 30
No	No	Yes	No	Yes	Yes	No	Yes	Yes	No

Test 31	Test 32	Test 33	Test 34	Test 35	Test 36	Test 37	Test 38	Test 39	Test 40
Yes	No	Yes	No	Yes	Yes	No	No	Yes	No

Test 41	Test 42	Test 43	Test 44	Test 45	Test 46	Test 47	Test 48	Test 49	Test 50
Yes	No	No	Yes	Yes	Yes	No	No	Yes	No

From the experiment, I found that 28 times out of the 50 trials there was a repeated birthday. This is slightly higher than expected at 56%. When consulting the data collected from the high school students, there were 25 pairs of birthday matches, 12 instances where 3 people had the same birthday, and 2 instances where 4 people had the same birthday. July 8 and August 14 must be a popular celebration day at my school. Clearly, the principle by von Mises is confirmed from this analysis of 186 students and a 50 trial run.

Interestingly, it was found that during the 2014 Fifa World Cup, the particular club teams had many pairs of birthdays. Conveniently, each of the 32 competing teams had a roster of 23 players. From the roster list, it was found that 16 of those clubs had 2 players who shared a birthday. Seems odd from first thought, but again, this 50% phenomenon reoccurs.

To go further, a study by Michael P. McDonald and Justin Levitt found how often people in a room are suspected to find a match to their exact birthdate: month, day, and year. Using a complex computer program, and surveying an age range of 18-81, they found that 180 people are needed to make these odds over 50%. In a group of 461 people, they concluded that a match would occur 99% of the time.


The graph depicts the findings of McDonald and Levitt to see how often people share an exact birthdate.

In the end, I believe I thoroughly solved the birthday problem, a philosophy that I wanted to learn.

So, the chances that you and I share a birthday if we are in a group of 23 people are somewhat higher than you probably thought.

A picture of me and my August 30th birthday twin, Mr. Warren Buffet:


-  By: Michael S., number enthusiast, saving the world one birthday at a time

For another description of the birthday problem, check out this TedEd video:

Sources:
https://medium.com/i-math/the-birthday-problem-307f31a9ac6f
https://pballew.blogspot.com/2011/01/who-created-birthday-problem-and-even.html
https://ed.ted.com/lessons/check-your-intuition-the-birthday-problem-david-knuffke
file:///Users/shoichem/Downloads/SSRN-id997888.pdf
https://www.theguardian.com/science/alexs-adventures-in-numberland/2014/jun/10/world-cup-birthday-paradox-footballers-born-on-the-same-day