The first college I went to had a 2-day long freshman orientation. Since we had to stay over night, I was randomly paired with a girl named Amanda. My orientation roommate did not talk very much, but I remembered her as being kind. I ended up having class with her and we sat next to each other. We began talking about how our birthdays were coming up. Turns out, we both had the same birthday, September 22nd. Coincidence? I think not.
Our orientation group was a few hundred students. There only needed to be 60 people in order for a 99.4% of 2 people sharing a birthday. Out of 23 people, there is a 50.7% chance that 2 of them share a birthday. How could this be? 23 people is a small number after all. This math formula is nicknamed the birthday paradox. The birthday paradox assumes that all 365 days (excluding leap day) all have an equal chance of being someone’s birthday.
You have a 365/365 or 100% chance of your birthday being your birthday. The next stranger you meet has a 364/365 or 99.7% chance of NOT sharing your birthday. Then, the third stranger you meet has a 363/365 or 99.5% chance of also not sharing your birthday. In order to figure out the chance of them NOT sharing a birthday, you multiply (365/365) * (364/365) * (364/365) to equal 99.2% of them not having the same birthday.
With every new stranger that you meet, the chance of them having a unique birthday decreases by one. Thus, the 4th stranger has a 362/365 or 99.2% chance of not having your birthday. By the 23rd stranger, they have a 343/365 or 94% chance of having not the same birthday as you. But, you need to compare the 23rd person to all of the strangers in the room. That is why you multiply every stranger's chance of having your birthday. A room of 23 people creates 253 possible comparisons to one another.
Thus, the complete birthday paradox formula for 23 strangers is:
(365/365) * (364/365) * (363/365) * (362/365) * (361/365) * … * (343/365)=49.3%
But that only shows the chances of 2 strangers NOT sharing the same birthday, thus, you must subtract 49.3% from 100%. This is how you end up with a 50.7% of 2 strangers sharing the same birthday. In order to test the birthday paradox with more strangers, you subtract the numerator by 1 and multiply the new fraction to the previous ones.
This diagram shows how the probability of sharing the same birthday with someone increases with more people in the equation.
The formula 1-(364/365)N=P can help you figure out how many people you need (N) for a chance that one person shares your birthday (P). For example, you need 840 people (N) for a 90% probability (P) that 1 person shares your birthday. As (N) increases, so does your probability.
Another fun fact about birthday probability is that you need 21,535 Facebook friends in order to have a 90% chance of every possible birthday of the year filled out! Also, find out which celebrity shares your birthday with you! Maybe you’re like me and shares a birthday with Tom Felton. Let me know in the comments below who your Birthday Twin is! I also tested the birthday paradox with random strangers I found at Montclair State University. See my results here!
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