Math Explains Likely Long Shots, Miracles and Winning the Lottery – Scientific American

#Math Explains Likely Long Shots: Nice illustration of the
combinatorics of why 23 people usually share a birthday


…because that’s the probability that none of them share my birthday, the probability that at least one of them has the same birthday as me is just 1 – 0.94. (This follows by reasoning that either someone has the same birthday as me or that no one has the same birthday as me, so the probabilities of these two events must add up to 1.) Now, 1 – 0.94 = 0.06. That’s very small.

Yet this is the wrong calculation to consider because that
probability–the probability that someone has the same birthday as you–is not what the question asked. It asked about the probability that any two people in the same room have the same birthday as each other. This includes the probability that one of the others has the same birthday as you, which is what I calculated above, but it also includes the probability that two or more of the other people share the same birthday, different from yours.

This is where the combinations kick in. Whereas there are only n – 1 people who might share the same birthday as you, there are a total of n × (n – 1)/2 pairs of people in the room. This number of pairs grows rapidly as n gets larger. When nequals 23, it’s 253, which is more than 10 times as large as n – 1 = 22. That is, if there are 23 people in the room, there are 253 possible pairs of people but only 22 pairs that include you.



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