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Happy Birthday (3) (Posted on 2006-06-09) Difficulty: 3 of 5
In Happy Birthday, the question was if there are N people in a room, what is the probability that there are at least two people in the room who share a birthday?

What if instead exactly two was required? If there are N people in a room, what is the probability that there are exactly two people in the room who share a birthday?

(Note: Assume leap year doesn't exist, and the birthdays are randomly distributed throughout the year.)

No Solution Yet Submitted by Sir Percivale    
Rating: 4.0000 (3 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution solution | Comment 3 of 12 |

The probability that the first two have the same birthday, but the rest do not match any others is:

(1/365) * (364*363*...*(367-n)) / 365^(n-2)

(verify that for n=4, the last factor shown is 367-4=363 so there are 2 factors in the numerator.)

This has to be multiplied by C(n,2) as any 2 could match, not just the first two.  Doing that, and simplifying:

(n!/(2*(n-2)!)  *  1/365  *  (364!/(366-n)!)/365^(n-2)

= n! * 364! / (2 * (n-2)! * 365^(n-1) * (366-n)!)

2       0.0027397260273972602
3       0.0081966597860761868
4       0.0163034931635378402
5       0.0269491530831082107
6       0.039980729847953688
7       0.0552062680640237227
8       0.0723983570227744893
9       0.091298436918739493
10      0.1116217191095547911
11      0.1330626033586047373
12      0.1553004630979879948
13      0.178005661563372541
14      0.2008456574078052734
15      0.2234910581903502305
16      0.2456214839133085899
17      0.266931110371540842
18      0.2871337731770581455
19      0.3059675275176000738
20      0.3231985754904329547
21      0.338624491638570418
22      0.352076697470787599
23      0.3634221566065063905
24      0.3725642830865206235
25      0.3794430756866886636
26      0.3840345101641759922
27      0.3863492387185902037
28      0.3864306611082042374
29      0.38435244446297493
30      0.3802155786223753955
31      0.3741450606453152687
32      0.3662863059468291946
33      0.3568013843699470953
34      0.3458651775544983743
35      0.3336615494548170286
36      0.3203796150929572694
37      0.3062101839753431123
38      0.2913424444489908364
39      0.2759609440497216422
40      0.2602429090245392272
41      0.2443559341076214478
42      0.2284560616828104632
43      0.2126862580263892665
44      0.1971752836967413083
45      0.18203694557284368
46      0.167369709734035478
47      0.153256647452048316
48      0.1397656811069543665
49      0.1269500928597238671
50      0.1148492563827981331
51      0.1034895507892186029
52      0.0928854159960219571
53      0.0830405099653933607
54      0.0739489304233444985
55      0.0655964665838506898
56      0.0579618499251315887
57      0.0510179769951220509
58      0.0447330813946466437
59      0.0390718363423945594
60      0.033996373425928568
61      0.0294672071682555212
62      0.0254440587923156121
63      0.0218865759708254482
64      0.0187549483529053067
65      0.0160104212292903314
66      0.0136157118211079505
67      0.0115353343563443543
68      0.0097358413504854051
69      0.0081859893620397015
70      0.0068568379869784607
71      0.0057217910317387068
72      0.0047565887089679112
73      0.0039392593815114532
74      0.0032500388854616183
75      0.0026712648373657136
76      0.0021872526154453951
77      0.0017841589370920794
78      0.0014498381672649478
79      0.0011736957118524139
80      0.0009465420984942544
81      0.0007604506417103484
82      0.0006086209416702343
83      0.0004852498845611623
84      0.0003854113015632459
85      0.0003049450040756231
86      0.0002403555436559527
87      0.0001887207427384047
88      0.00014760980171116
89      0.0001150106034935449
90      0.0000892657019020571
91      0.0000690163881475163
92      0.0000531541741288329
93      0.000040779005423782
94      0.0000311635151037848
95      0.0000237226463479688
96      0.0000179880025470345
97      0.0000135863240578755
98      0.0000102215375232224
99      0.00000765987384737
100     0.0000057176022288169
101     0.0000042509783168505
102     0.0000031480532576567
103     0.0000023220361910267
104     0.0000017059450415515
105     0.0000012483188746663
106     0.0000009097995254615
107     0.0000006604206861927
108     0.0000004774693056134
109     0.0000003438072413893
110     0.0000002465609009152
111     0.000000176103419892
112     0.0000001252680864786
113     0.000000088743542345
114     0.0000000626110872875
115     0.0000000439924671982
116     0.0000000307830972108
117     0.0000000214509968652
118     0.0000000148859988165
119     0.000000010287216885
120     0.0000000070794777913

  Posted by Charlie on 2006-06-09 11:50:07
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