Birthday Paradox

 Have you ever wondered how many people must gather in a room in order for you to find your birthday twin? Probability theory says it's merely 23. Yes! There is a 50% chance (50.73% to be precise) that at least 2 individuals out of 23  will share their birthday. Move the party to a bigger room to accommodate 75 people and you will find out that there is a 99% chance that at least 2 people will share their birthday! 

As strange as it sounds, it is in fact true. How do we get it?

Let's take an example to understand this better. Assume two friends, Rahul and Anjali. The probability of Rahul and Anjali having different birthdays is 364/365, i.e, around 99.7%! Add Tina to the group and the probability of all of them having unique birthdays falls down to 363/365. Aman joins the group and the probability of an unrepeated birthday is further reduced  to 362/365. We continue doing this till we reach the 23rd person. The probability of the 23 individuals having a unique birthday is 343/365.

We multiply all these terms together and find out the probability that no one shares their birthday out of these 23 is 49.27%

Since the odds of a matching birthday and odds of a unique (not-matching)  birthday always add up to 100% , we can simply subtract this figure from 100, we thus arrive at 50.73%!!

The reason behind this more-than-even (i.e, >50%) chance in such a small group is the number of possible pairs that can be created here. In a group of 23, 253 pairs can be created (such as Tina-Anjali, Tina-Rahul, Tina-Aman, Tina-Mrs Briganza, Anjali-Anjali, Anjali-Rahul, and so on). 

As the size of the group grows, the number increases quadratically, however, our brains aren't good at grasping non-linear functions, thus this situation doesn't seem intuitive to us! This is just one of the many ways in which probability theory proves our intuition wrong.

Move the party to a bigger room to accommodate 75 people and you will find out that there is a 99% chance that at least 2 people will share their birthday!

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