Skip to main content

Birthday Paradox



Why not delve into some serious maths?  The 'Birthday Paradox' is actually not a paradox (a statement which is true and false at the same time) but it is a statement that is too interesting that it seems false to our human minds.  

What is the probability that two persons will have the same birthday? There is a 1 in 365 chance for this to happen or a probability of 1/365. Then I can say the probability of 2 people not having the same birthday is 

Hence, there's a 99% chance of 2 people having different birthdays.

But if instead of 2 there were 23 people, then what would be the probability of 2 people having the same birthday. For this, we must consider all the possible pairings between these people and it is equal to

There are a total of 253 possible pairings between a group of 23 people. Since each pair has a probability of 364/365 of having different birthdays, the probability of finding a pair among 253 possible pairs who have different birthdays will be equal to 

Then the probability of finding a pair who has the same birthday is equal to  

This means there is a 50% chance of two people having the same birthday in a group of 23 people. This is the birthday paradox. The probability just skyrocketed from 1% to 50% by just adding 21 people to the group.

To make this further interesting, let us consider a group of 50 people, then the probability of finding a pair with the same birthday will be equal to 

Imagine a room with 50 people, there is a 97% chance that two of them will have the same birthday, which is insane! Math never fails to surprise us.



Comments

Popular posts from this blog

Prime Numbers in Nature

 Prime numbers are one of the easiest concepts to understand and learn in Maths. But some of the complex problems in Maths belong to this branch. Never would have imagined an application of prime numbers in nature. Thanks to the book 'Weird Maths: At The Edge of Infinity and Beyond' which introduced me to this along with other weird applications of Maths. There's a particular species type of cicadas found in the US called the Magicicada or the 'periodical cicadas' who stay in their larva for 13 or 17 years. Imagine staying inside for over a decade with no connection to the outside world and coming out in the year 2021. People wearing masks walking all around you, smartphones, oh, the shock!🤯 It is the primality of these numbers which interest the mathematicians. The reasons are quite interesting. As the lifecycle of the prey changes, the population cycle (pattern in the population for a period of time) of their predators also vary due to the change in the availabil...

Not a Typical Movie Review

 We were assigned with the task of watching a movie and understand its educational implications. I had  chosen the movie 'Dead Poets Society (1989)' directed by Peter Weir starring the late Robin Williams. I'd seen this movie a while ago but this time, I decided to watch it to understand what makes John Keating different from other teachers. This Academy award winning movie is set in a conservative boy's boarding school named Welton's Academy which is known for churning out disciplined and academically brilliant students. The story focuses on a group of seventeen year old boys and their English teacher John Keating (Robin Williams). John Keating is a nonconformist and lives by the phrase 'Carpe Diem' which means 'seize the day'. He encourages his students to get rid of their fears, find their voices and live their lives to the fullest through poetry. Their newfound interest in poetry led them to create a secret club named Dead Poets Society. One of t...

The Story of Probability

 Why was the concept of probability invented? This is the question we are going to answer through this post. Probability was discovered to answer a gambling problem. The original problem is a bit complicated but the simplified version goes like this.  Suppose two persons, A and B are playing a game of coin toss where the person who gets their toss thrice first wins a pot of gold. If the score is 2-1 and the game has to be stopped due to some unforeseeable reasons then how would the pot of gold be divided?  By logic, since A is leading with 2 points and B only has 1 point, the pot can be divided in the ratio of 2:1 . This seems like a good argument but this fails when the score is 0-1. In this case, it seems unfair to give the pot of gold to B based on the previous argument when A may have a chance to win in the upcoming tosses. But if one were to calculate the probability of winning the game of the two players, then it would give us a foolproof explanation as to how...