Leap Day Birthday Math

Happy leap day!!!

Here are some leap-day birthday thoughts I discussed with my colleagues and students today:

What’s the probability of a leap year birthday?

The probability that someone is born on a leap day is \frac{1}{365\cdot 4+1}=\frac{1}{1461}\approx 0.000684. Oh wait, that’s not completely true. Leap years don’t really occur every four years. Years divisible by 100 are not leap years, unless also divisible by 400. So, the actual probability is

\frac{100-4+1}{365\cdot 400+100-4+1}= \frac{97}{146097}\approx 0.000639.

What’s the probability of having triplets on a leap day?

One of our RM students is a triplet, born today. What are the chances of that occurring? Well, the statistics on triplets are pretty hard to get right. But let’s say the occurrence of a triplet birth is 1 in 8000. (That’s my informal estimate based on this site and this site.) I think we can say that the probability of being a triplet is 3 times that (right?). Then, the probability of being a triplet born on a leap day is

\left(\frac{100-4+1}{365\cdot 400+100-4+1}\right)\left(\frac{3}{8000}\right)= \frac{291}{1168776000}\approx\frac{1}{4016412} \approx 0.249 \times 10^{-7}.

The current US population is 311,591,917, so that means there are roughly 77 triplets in the US with leap day birthdays. Happy birthday to all of you!

Bonus thought question: Iif you have quadruplets born on a leap day, you get to celebrate 4 birthdays every four years, so doesn’t that average out to one birthday a year?

Half-birthday for those born on August 29

One of my other colleagues has a birthday on August 29th. So today is her half birthday! But it only comes around every four years (roughly). Hooray!

But then that got us thinking about half birthdays: Some people, like those born on August 30th or 31st NEVER have a half birthday. How sad!! This happens to anyone born on August 30th, August 31st, March 31st, October 31st, May 31st, or December 31st. That’s a lot of people without half birthdays.

But wait. When is your actual half birthday? Shouldn’t it be 182.5 days before/after your birthday? That’s not necessarily the same date in the month. For instance, my birthday is May 15. So my half birthday should be November 15, right? Wrong. My half birthday is (May 15 + 182.5 days), which is November 13th or November 14th, depending on if you round up or down. Even accounting for a leap year, it’s still not quite right.

Who else is miscalculating their half birthday? Unless your birthday is in June, April, October, or December, you’re half-birthday isn’t what you think it is. To calculate your half birthday, use this amazing half birthday calculator I just discovered!

2 thoughts on “Leap Day Birthday Math

  1. Mr. Chase,
    Interesting web page; so glad I found it as we have been wondering this for 8 years. It gets a little muddled when you start thinking of all those planned inductions and c-sections on leap day. We have a set of spontaneous (the 1 in 8,100 kind) triplets born on Leap Day not planned! Not only did they arrive on Leap Day (a Sun. night in 2004) without a planned entrance on that day, they were born naturally, not via C-section. Here is a link to an article that ran in one paper today (they were in several). We think they are the only spontaneous, non-induced naturally born triplets born on Leap Day in the world.
    Jeff and Kelly Rowe


  2. Another thought I don’t see here…
    Birthdays aren’t spread out evenly throughout the year. There are peaks and valleys. More babies are born in late summer and autumn because of being closed up in winter … if there is a power outage, you’ll see a spike in births about nine months later…

    I don’t know the exact math… but it’s not quite even.
    Purely statistically, you’re right – but the 1/1461 (or 1/146097) doesn’t quite hold up to reality.

    Realistically, the odds are a bit less… not standard deviations off, but … less. 🙂

    btw… for those who don’t quite get the math, there are 365.2425 days in the average year…
    365 days in the average year
    Plus one every four years (1976, 1980, 1984, …)
    Except every 100th year (1700, 1800, 1900)
    Except every 400th year (2000)

    So that’s where we get 365 + (1/4) – (1/100) + (1/400) = 365 + .25 – .01 + .0025 = 365.2425
    In 400 years, there are 146097 days, 99 of which are leap year days.

    So the odds of being born on any given day, purely statistically, would be 400/146097
    Approximately 0.0027379
    (400/146097)*365 = ~0.9993361
    So you’ve got a 99.93361% chance you were *not* born on February 29th. 🙂

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