The Birthday Surprise
---------------------

Problem

What is the least number of people n such that with probability >= 1/2 
two will have the same birthday?  

Assume that people have random birthdays and that all years have 365 days.


Solution

Pr[2 of n people have the same birthday] 
 = 1 - Pr[all n people have different birthdays] 
 = 1 - Pr[person 2 has a different birthday than person 1   &&
          person 3 has a different birthday than persons 1 and 2 &&
          person 4 has a different birthday than persons 1, 2, and 3  &&  ... &&
          ...
          person n has a different birthday than persons 1, 2, ..., n-1 ]
 = // handwave -- I'd need to develop things a bit more to justify this 
   1 - (1-1/356)*(1-2/365)*...*(1-(n-1)/356) 

So now we just calculate:

// Birthday problem

#include <math.h> 
#include <iostream.h>

void main() 
{    
  double  prod = 1;

  for (int i=0; i<59; i++) {
    prod *= 1 - i/365.0;
    cout << i+1 << " persons: probability is " << 1-prod << endl;
  }
}

Which give output:

1 persons: probability is 0
2 persons: probability is 0.00273973
3 persons: probability is 0.00820417
4 persons: probability is 0.0163559
5 persons: probability is 0.0271356
6 persons: probability is 0.0404625
7 persons: probability is 0.0562357
8 persons: probability is 0.0743353
9 persons: probability is 0.0946238
10 persons: probability is 0.116948
11 persons: probability is 0.141141
12 persons: probability is 0.167025
13 persons: probability is 0.19441
14 persons: probability is 0.223103
15 persons: probability is 0.252901
16 persons: probability is 0.283604
17 persons: probability is 0.315008
18 persons: probability is 0.346911
19 persons: probability is 0.379119
20 persons: probability is 0.411438
21 persons: probability is 0.443688
22 persons: probability is 0.475695
23 persons: probability is 0.507297   <<<< first number exceeding 0.5
24 persons: probability is 0.538344
25 persons: probability is 0.5687
26 persons: probability is 0.598241
27 persons: probability is 0.626859
28 persons: probability is 0.654461
29 persons: probability is 0.680969
30 persons: probability is 0.706316
31 persons: probability is 0.730455
32 persons: probability is 0.753348
33 persons: probability is 0.774972
34 persons: probability is 0.795317
35 persons: probability is 0.814383
36 persons: probability is 0.832182
37 persons: probability is 0.848734
38 persons: probability is 0.864068
39 persons: probability is 0.87822
40 persons: probability is 0.891232
41 persons: probability is 0.903152
42 persons: probability is 0.91403
43 persons: probability is 0.923923
44 persons: probability is 0.932885
45 persons: probability is 0.940976
46 persons: probability is 0.948253
47 persons: probability is 0.954774
48 persons: probability is 0.960598
49 persons: probability is 0.96578
50 persons: probability is 0.970374
51 persons: probability is 0.974432
52 persons: probability is 0.978005
53 persons: probability is 0.981138
54 persons: probability is 0.983877
55 persons: probability is 0.986262
56 persons: probability is 0.988332
57 persons: probability is 0.990122
58 persons: probability is 0.991665
59 persons: probability is 0.992989
60 persons: probability is 0.994123


So the answer is 23 people.  
Many people find that number surprisingly low, and therefore
call this the birthday "paradox".
But I think there is no paradox here.