Birthday Paradox

In a group of 23 people, the probability that two or more people will share their birthdays is greater than 50%. For a group of 57 people, this probability increases to 99%.

This may seem counter-intuitive at first sight. There are 365 possible days on which a birthday can occur. So it may seem odd that in a group of 57, there is 99% probability of a shared birthday.

Question: In a group of ‘n’ people, what is the probability that any of those ‘n’ persons shares a birthday with any of the others in the group?

This is very simple.

The answer is simply:

1 – (364/365) (363/365) … ((365 – n + 1 )/365 )

=   365 !  / [ (365^n) (365 – n)! ].

Approximating this function, and plugging in values of n, we get the probability of a group of those many people having a common birthday in their midst.

See http://en.wikipedia.org/wiki/Birthday_paradox for more on this.

2 Responses to Birthday Paradox

  1. Sharadh says:

    Ya you are right about the birthday paradox. I have read about this before, and at first sight it also seemed counter intuitive to me.🙂

    Your voter paradox was good, pardon me for not commenting there (wanted to club the comments together). I haven’t read about it before.

    One small clarification: My blog doesn’t deal particularly with dharma (saw your categorization) even though I refer to the Bhagavat-Gita very often. I’m not telling you to change the categorization; just pointing out my opinion on my own blog.🙂

    Will read your thesis sometime later, whenever I have time. Till then, all the best.

  2. tkramesh says:

    Hey, thanks for reading and commenting on this. Many such interesting topics are given at http://www.math.ucla.edu/~tao/puzzles.html.

    You are right about your blog. I think I shall change the categorization to “Others”. Thanks for pointing it out.

    You’ve been writing really good posts on your blog. Keep it up!🙂

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