birthday - The Birthday Paradox September 18, 2003 1 Two...

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The Birthday Paradox September 18, 2003 1 Two Birthdays the Same It turns out that there is at least a 50% chance that in any random sample of 23 people, two of them will have the same birthday. By “birthday”, I mean you don’t include the year; so, an example of a birthday would be “June 11”. The reason that one uses the “paradox” to refer to this phenomenon is that it seems counterintuitive that a random sample of so few people should likely have a matching pair of birthdays. The reason that such a small number of people suFces is that there are many pairs of individuals, and so many chances for a collision of birthdays: Indeed, with 23 people, there are ( 23 2 ) = 253 pairs of individuals. Let us now prove that the probability is indeed 50%. ±irst, let the sample space S be the set of all sequences ( x 1 , ..., x 23 ), where 1 x i 365. The value of x i indicates the day of the year of the i th person’s birthday. We are assuming here, for simplicity, that there are no leap years, so that each year
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birthday - The Birthday Paradox September 18, 2003 1 Two...

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