# Happy Birthday To Me!

by Kevin J. Lin

With one human there is a 0 percent chance that you'll have two humans
with the same birthday.

With two humans the probability that they *won't* share a birthday
is 364/365. The probability that they *will* share a birthday is
therefore 1 - (364/365).

With three humans the probability that they won't share a birthday is
the same as for two humans, times 363/365. So the probability that three
humans will share a birthday is 1 - (364/365) * (363/365). Notice that
with each additional person added, the probability that he or she shares
a birthday with one of the previous persons goes up, because there are
fewer "free" days remaining.

Following this progression, the probabilities are:

Number of Humans |
Probability of two shared birthdays |

1 |
0 |

2 |
0.00273972602739725 |

3 |
0.00820416588478134 |

4 |
0.0163559124665502 |

5 |
0.0271355736997935 |

6 |
0.0404624836491114 |

7 |
0.0562357030959754 |

8 |
0.074335292351669 |

9 |
0.0946238338891667 |

10 |
0.116948177711078 |

11 |
0.141141378321733 |

12 |
0.167024788838064 |

13 |
0.194410275232429 |

14 |
0.223102512004973 |

15 |
0.252901319763686 |

16 |
0.28360400525285 |

17 |
0.315007665296561 |

18 |
0.34691141787179 |

19 |
0.379118526031537 |

20 |
0.41143838358058 |

21 |
0.443688335165206 |

22 |
0.47569530766255 |

23 |
0.507297234323986 |

So 23 humans will have a better than average chance of sharing a birthday.

What about our misunderstood friends the Martians, who have a year of
nearly 670 days? (As someone in the discussion forums pointed out, they
may have leap days as well, but it turns out not to matter.)

Surprisingly, only eight more Martians will be needed.