Birthday Paradox

Something amazing paradoxical given by probability which is hard to accept in real life.

I had to sit and analyze the proof to accept this:

“In a room of just 23 people there’s a 50-50 chance of two people having the same birthday. In a room of 57 there’s a 99% chance of two people matching.”

Here is the explaination:  (reference: http://mathworld.wolfram.com/)

Consider the probability that no two people out of a group of will have matching birthdays out of equally possible birthdays. Start with an arbitrary person’s birthday, then note that the probability that the second person’s birthday is different is , that the third person’s birthday is different from the first two is , and so on, up through the th person. Explicitly,

so the probability that two or more people out of a group of do have the same birthday is therefore

If 365-day years have been assumed, the number of people needed for there to be at least a 50% chance that at least two share birthdays is the smallest such that . This is given by , since

Hence the astounding Birthday Paradox..