Birthday distribution: Difference between revisions
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#REDIRECT [[birthday problem]] |
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Assume N people are in a room. What is the smallest N such that there |
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is at least probability 0.5 that M people have the same birthday? Assume |
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the birthday distribution is uniform over 365 days. |
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Assume X~Mult(N;P) on k=365 categories, with P=(1/365,...,1/365) and |
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calculate Prob(M;N) = P[max X[i] is at least M] using the exact method of |
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B. Levin, "A representation for multinomial cumulative distribution functions", |
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Annals of Statistics, 9:1123-1126 (1981). |
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M N Prob(M;N) |
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----- ----- --------- |
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1 1 1 |
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1 0 0 |
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2 23 0.507297234324 |
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2 22 0.475695307663 |
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3 88 0.511065110625 |
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3 87 0.499454850632 |
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4 187 0.502685373189 |
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4 186 0.495825706383 |
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5 313 0.501070475849 |
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5 312 0.496195571644 |
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6 460 0.50244941037 |
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6 459 0.498637523705 |
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7 623 0.502948948663 |
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7 622 0.499795687887 |
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8 798 0.500320275207 |
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8 797 0.497616714463 |
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9 985 0.500948416378 |
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9 984 0.498568067031 |
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10 1181 0.500931161057 |
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10 1180 0.498796176416 |
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11 1385 0.501001958747 |
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11 1384 0.499059528126 |
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12 1596 0.501166734721 |
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12 1595 0.499379622274 |
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13 1813 0.501104224633 |
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13 1812 0.499445407288 |
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14 2035 0.500348626509 |
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14 2034 0.498798065243 |
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15 2263 0.501295448239 |
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15 2262 0.499836197505 |
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I am grateful to Professor Bruce Levin for providing this table. |
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Postscript (October, 1998) |
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Haikin Fedor has returned to this problem and calculated N up to M=18. The required computation times on his Sparc20 are also indicated. Are more efficient exact algorithms possible? |
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*** M= 2 implies N= 23 *** Thu Sep 17 16:03:09 1998 |
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*** M= 3 implies N= 88 *** Thu Sep 17 16:03:15 1998 |
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*** M= 4 implies N= 187 *** Thu Sep 17 16:03:24 1998 |
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*** M= 5 implies N= 313 *** Thu Sep 17 16:03:50 1998 |
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*** M= 6 implies N= 460 *** Thu Sep 17 16:05:51 1998 |
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*** M= 7 implies N= 623 *** Thu Sep 17 16:10:26 1998 |
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*** M= 8 implies N= 798 *** Thu Sep 17 16:19:15 1998 |
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*** M= 9 implies N= 985 *** Thu Sep 17 16:34:01 1998 |
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*** M= 10 implies N= 1181 *** Thu Sep 17 16:57:08 1998 |
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*** M= 11 implies N= 1385 *** Thu Sep 17 17:29:46 1998 |
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*** M= 12 implies N= 1596 *** Thu Sep 17 18:13:54 1998 |
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*** M= 13 implies N= 1813 *** Thu Sep 17 19:11:39 1998 |
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*** M= 14 implies N= 2035 *** Thu Sep 17 20:27:26 1998 |
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*** M= 15 implies N= 2263 *** Thu Sep 17 22:11:58 1998 |
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*** M= 16 implies N= 2494 *** Fri Sep 18 01:52:27 1998 |
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*** M= 17 implies N= 2731 *** Fri Sep 18 07:30:00 1998 |
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*** M= 18 implies N= 3015 *** Fri Sep 18 16:22:11 1998 |
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[[Category:Probability distributions]] |
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Latest revision as of 02:05, 21 October 2007
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