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Detail of a portrait of Dr. Johnson by Joshua Reynolds (1775)

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John Derbyshire

So the chance of all N birthdays being different is 365/366 × 364/366 × 363/366 ×  . . .  × (367 − N) / 366.

For N = 2 to 40, the actual values of this expression, to three decimal places, are as follows: 0.997, 0.992, 0.984, 0.973, 0.960, 0.944, 0.926, 0.906, 0.883, 0.859, 0.833, 0.806, 0.777, 0.748, 0.717, 0.686, 0.654, 0.622, 0.589, 0.557, 0.525, 0.494, 0.463, 0.432, 0.403, 0.374, 0.347, 0.320, 0.295, 0.271, 0.248, 0.226, 0.206, 0.187, 0.169, 0.152, 0.137, 0.123, and 0.109.

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So in fact the chance that all the birthdays are different after you’ve stopped 40 people is close to one in ten. To put it the other way round, the chance that at least two of the birthdays are the same is almost 90 percent.

The 50-50 point actually falls between N = 22 and N = 23.

(The math needs tweaking there to account for the fact that one particular birthday — February 29, of course — does not occur one time in 366, but only one time in 1,461. As math textbooks say, I leave that as an exercise for the reader. The general principle is plain enough.)

Now a book recommendation. If you like exploring math and feel up to 248 pages of exceptionally lucid exposition, Avner Ash and Robert Gross’s Elliptic Tales is just the ticket. It’s basically an exposition of the cumbersomely named Birch and Swinnerton-Dyer Conjecture, one of the great unsolved problems in math, but along the way you learn a lot of good stuff about algebra, number theory, and analysis. Strongly recommended.



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