# Chance of sharing a birthday

## Clinical bottom line

With 40 people in a room, there is a 90% chance that any two will share a birthday.

Even with 365 people in a room, there is only a chance of just below 1 in 2 that any two will share a particular birthday.

## Data sources

Your brain.

## What the sources tell us

- There are 365 days in a year. Only one day out of those 365 is a birthday.
- Therefore the chance of anyone having a birthday on a particular day is 1 in 365.

### Give us the odds

The chance of:

- two people sharing a birthday would be 1 - (364/365), or 0.3%, or 1 in 370
- three people sharing a birthday would be 1 - ((364/365)(363/365)), or 0.8%, or 1 in 122
- four people sharing a birthday would be 1 - ((364/365) )(363/365)(362/365)), or 1.6%, or 1 in 61, and so on.

The table gives the approximate odds. It rapidly becomes highly probable that two people will share a birthday when the number of people present is 40 or more.

The situation is different if you want to find the chance of someone sharing a particular birthday, when many more people are needed. The difference is that in the example of any birthday, the number of available days reduces. With a particular birthday, it is always 364/365. So instead of chances of 0.3%, 0.8%, and 1.6% with 2, 3 or four people, the chance becomes 0.3%, 0.5%, and 0.8% - with the gap rapidly growing.

So for the chance of two people sharing a particular birthday, with 25 people it is only 6%, with 100 people about 24%, with 200 people about 42%, and even with 365 people it is only about 60%

### Table: Chance of two people sharing a birthday

Number of people |
Probability |
Chance % |
Approximately 1 in |

2 |
0.0027 |
0.3 |
370 |

3 |
0.0082 |
1 |
122 |

4 |
0.0164 |
2 |
61 |

5 |
0.0271 |
3 |
37 |

10 |
0.117 |
10 |
8.5 |

15 |
0.2529 |
25 |
4.0 |

20 |
0.4114 |
40 |
2.4 |

25 |
0.5687 |
60 |
1.8 |

30 |
0.7063 |
70 |
1.4 |

35 |
0.8144 |
80 |
1.2 |

40 |
0.9032 |
90 |
1.1 |

45 |
0.9483 |
95 |
1.1 |

50 |
0.9704 |
97 |
1.0 |

55 |
0.9863 |
99 |
1.0 |

60 |
0.9941 |
99.5 |
1.0 |

## Comment

As always with risk and chance, the answer you get is about how you ask the question. Always listen carefully, and think.