“There are three hundred and sixty-four days when you might get un-birthday presents, and only one for birthday presents, you know.” -Lewis Carroll

It’s the end of the week once again, and so it’s time for yet another Ask Ethan column! This week, I was asked a wonderful math question by Keith H., who wanted to know:

Among my 1,434 [Facebook] friends, it’s nobody’s birthday today. Given the 365.25 days in a year, what are the odds of a day like this? Show your work.

Assuming that none of your friends were born prior to March 1, 1900, this is the right starting point.

In the last 114 years, there has been a leap year every four years. If you assume that everyone you know was born at a completely random date-and-year in that timeframe (a generally good *enough* assumption), there’s a 4-out-of-1461 chance they were born on any given date *except* for February 29th, and a 1-out-of-1461 chance that they were, in fact, born on February 29th.

Assuming you had *any number* of friends, what are the odds that **no one** would have a birthday on today’s date?

Let’s start with one person.

Assuming you have just *one* friend and that today is a non-leap day, the odds that this friend of yours will not have a birthday today is 1457-out-of-1461, or **99.73%**. The nice thing is, this is true *for every single friend you have*, independently.

So the second friend has a 99.73% chance of not having today be their birthday, the third friend has a 99.73% chance of today not being their birthday, the 100th friend has a 99.73% of today not being their birthday, and so on. Let’s say you have six friends, to keep things simple: what are the odds that *no one’s* birthday is today?

It’s 98.37%, because you need the first person to not have their birthday today **and** the second person to not have their birthday today **and** … and so on. If *anyone* has their birthday today, the game is ruined.

The fun thing is, you can extend this for an arbitrary number of people! Here are some fun stats.

Number of Friends | Each one’s unbirthday odds | Total unbirthday odds | Expressed in percent |
---|---|---|---|

1 friend | 0.99726215 | (0.99726215)^1 | 99.73% |

2 friends | 0.99726215 | (0.99726215)^2 | 99.45% |

5 friends | 0.99726215 | (0.99726215)^5 | 98.64% |

10 friends | 0.99726215 | (0.99726215)^10 | 97.30% |

50 friends | 0.99726215 | (0.99726215)^50 | 87.19% |

150 friends | 0.99726215 | (0.99726215)^150 | 66.28% |

253 friends | 0.99726215 | (0.99726215)^253 | 49.98% |

365 friends | 0.99726215 | (0.99726215)^365 | 36.76% |

500 friends | 0.99726215 | (0.99726215)^500 | 25.39% |

1000 friends | 0.99726215 | (0.99726215)^1000 | 6.447% |

1434 friends | 0.99726215 | (0.99726215)^1434 | 1.961% |

1680 friends | 0.99726215 | (0.99726215)^1680 | 0.9993% |

2000 friends | 0.99726215 | (0.99726215)^2000 | 0.4156% |

2500 friends | 0.99726215 | (0.99726215)^2500 | 0.1055% |

3000 friends | 0.99726215 | (0.99726215)^3000 | 0.02679% |

5000 friends | 0.99726215 | (0.99726215)^5000 | 0.0001113% |

(The caret symbol — ^ — denotes an exponential.)

As you can see, if you have only a few friends, there are very good odds that on any particular day, no one will have a birthday. But those odds drop down to 50-50 at the 253 friend mark, and decrease rapidly from there. By time you get up to Keith’s number of 1434 friends, you’ve got less than a 2% chance of *no one* having a birthday today; those odds drop below 1% at 1680, and by time you get to Facebook’s limit — 5000 friends — there’s only a 0.0001113% chance of having friends without a birthday today, or about 1-in-900,000.

But what if we wanted to go more general? What if we wanted to know what the odds are of having **at least one day in the year where no one has a birthday?**

That’s a lot harder.

You see, whatever the odds are that *no one* has a birthday today, you subtract that from 100% to get the odds that *someone* has a birthday today. In order to have at least one day where no one has a birthday, you can subtract the odds that **someone has a birthday on every single day** from 100%. (Obviously, this is only possible if you have at least 365 friends!)

Here’s what the odds of that look like.

Number of Friends | Daily unbirthday odds | Daily birthday odds | Odds of birthday every day | Subtracted from 100% |
---|---|---|---|---|

365 friends | 36.76% | 63.24% | (0.6324)^365 | 100% – 2.3 × 10^-71% |

500 friends | 25.39% | 74.51% | (0.7451)^365 | 100% – 2.3 × 10^-45% |

1000 friends | 6.447% | 93.553% | (0.93553)^365 | 99.9999999973% |

1434 friends | 1.961% | 98.039% | (0.98039)^365 | 99.927% |

1680 friends | 0.9993% | 99.0007% | (0.990007)^365 | 99.927% |

2000 friends | 0.4156% | 99.5834% | (0.995834)^365 | 97.44% |

2500 friends | 0.1055% | 99.8945% | (0.998945)^365 | 31.97% |

3000 friends | 0.02679% | 99.97321% | (0.9997321)^365 | 9.32% |

5000 friends | 0.0001113% | 99.9998887% | (0.999998887)^365 | 0.0406% |

And there you have it: if you have fewer than about 2000 friends, there’s a *near certainty* that there will be at least one day where none of your friends has a birthday, and yet if you have more than about 4000, there’s virtually *no chance* that you’ll have such a day! Food for thought — and a little math fun — for this week’s Ask Ethan.

And now that we’ve reached the end of another Ask Ethan column, it’s time for me to flip the script and ask *you* a question! I know that a great many of you are continuing to enjoy all the great Starts With A Bang content on our new home on Medium, but I also know that most of you are dissatisfied with the options for commenting there.

So I thought I’d throw out an idea and see what you thought: what if I began posting here the introduction to each new post with a link to the full text — something like this Tumblr website I set up — where each post would look something like this:

Then we could have open comments here — and I can still moderate, etc. — so all of you could still have your say both to me and with one another. The Medium gig is too beneficial for me in a myriad of ways to stop doing it, but the one thing I miss the most is your participation. What do you think of this idea? I thought I’d put together a poll for your feedback, if you’d like to participate.

Feel free to comment below if you have suggestions, and I hope you enjoyed today’s Ask Ethan. Don’t forget to leave your questions and suggestions here!