# 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.

## What the sources tell us

1. There are 365 days in a year. Only one day out of those 365 is a birthday.
2. 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.