Birthday Paradox Question

Not a long time ago I got a very interesting question on The Birthday Paradox and I thought I will share it with you.

The question is as follows:

It’s almost a Mathematical certainty that in a group of 100 people at least 2 will have the same birthday, in my smallish group of friends (about 100) I have just noticed 4 pairs have the same birthday, Also one of the pairs has the same birthday as me. I know theoretically 367 people will definitely give us 1 pair, but what is the largest group among our friends WITHOUT 2 people sharing a common birthday?

With this question I thought I will say a couple of more things about The birthday Paradox. Everything about this is about probabilities, nothing more or less. To understand what I am talking about, here is an example: in a room of just 23 people there’s a 50-50 chance of two people having the same birthday and in a room of 75 there’s a 99.9% chance of two people matching. To understand more about how to calculate this, I recommend you read the article Understanding the Birthday Paradox. It ends with a very useful formula, which helps us understand more. Also, I have used that formula to try and get an answer for the above question. Here is the formula (with n = the number of people):

\displaystyle{p(n) = 1 - \left(\frac{364}{365}\right)^{C(n,2)} = 1 - \left(\frac{364}{365}\right)^{n(n-1)/2} }

My logic goes as follow. For example, in a group of just 23 people there’s a 50-50 chance of two people having the same birthday. Or you can say that there is a 50 – 50 chance of 2 people not having the same birthday. As the group gets smaller the probability decreases, such that in a group of 10 people the probability that 2 people have the same birthday is 12%. I think that because we talk about probability (%) there is not a precise answer.

Therefore, if we think about the largest group which has no coincidental birthdays, we want our probability to be 0, i.e. we want to find such that P(n) = 0.  If we take a look at the above formula, we observe that the 2nd term should be 1. Because that is an exponential function, we should know that the power should be 0 to get 1. Therefore, we go directly to the fact that n(n-1)/2 = 1, which gives us immediately n = 0 or n = 1. So I would say we need a group of just 1 person, because a group with 0 people is not a group anymore. Quite interesting that even in a group of 2 people there is a probability of 0.2% for them to have the same birthday.

I have to confess that I got quite excited about this question, but I have the feeling there should be more to it. If there is anyone that has any ideas let me know in the comment box bellow, I would love to know more.

If you want to read more about probability, I totally recommend the following 3 books: Probability: A Very Short Introduction (Very Short Introductions) by John Haigh, Probability: For the Enthusiastic Beginner by David J. Morin and Probability: An Introduction by Geoffrey Grimmett.
  

Hope you enjoyed this post and that you are excited for the future ones. Have a great week. You can find me on Facebook, Tumblr, Google+, Twitter  and Instagram. I will try to post there as often as possible.

Don’t forget that maths is everywhere! Enjoy! 

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