It’s Sunday… it means it’s time for “Just for Fun”. Every Sunday, we publish a posting not related to finance or investment.

Now, let’s play a little bit with probability.

If there are 23 people in a room, what is the probability that two persons share the same birthday?

Can you believe that the probability is more than 50%? Yes… I am not kidding, the answer is 50.7%. How come? There are 365 days in a year and there are only 23 people in a room.

I was very surprise too…. Although I am not an expert in probability, I will try to explain how we get this number.

First, let’s start with a simple problem. We have 3 people in a room. What is the probability that two persons share the same birthday? Let’s take a date for the first person. The second person has probability of 1/365 to share the same birthday (we ignore leap year for simplicity). The third person also has the same probability, 1/365. We end up with 2/365.

Do we miss something? Indeed, yes. There is a probability that the second person shares the same birthday as the third person. Things are getting more complicated for 23 people.

How do we solve this problem then? We all know this formula from our class in high school:

P(two persons share the same birthday) = 1 – P(no two persons share the same birthday).

It’s easier to find the probability of no two person share the same birthday, isn’t it?

Back to our problem, let’s take a date for the first person from 23 people. The probability that the second person does not share the same birthday is 364/365. The probability of the third person is 363/365. The probability of the fourth person is 362/365, and so on. So we end up in the following equation:

P(two persons share the same birthday) = 1 – (364 / 365) * (363 / 365) * … * (343 / 365)

P(two persons share the same birthday) = 50.7%

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