Careful With Those Odds

There's a story circulating today about a child born today, Leap Day, to a mother also born on Leap Day. This is presented as being a two-million-to-one event, and it's not hard to see where they get those numbers. Suppose you take a mother-child pair at random. What are the odds they'll both have been born on February 29th? Well, Leap Day occurs (roughly) once every four years, which adds up to once every 1461 days. So if one in 1461 is the odds of the mother's being born on Leap Day, and also the odds of the child's being born on Leap Day, then assuming the two are independent events the odds of the coincidence are 2,134,521-to-one.

But this isn't necessarily the right way to think about it. How surprised should I be that today, February 29th, 2012, featured a coincidence like this? Well, there are seven billion people in the world, and the global annual birth rate is roughly twenty births per thousand people. That's 140 million births in a year; in a leap year, that's 382,514 births per day on average, but let's be conservative. Let's be really conservative, actually, and say there will only be 200,000 births today. That's really conservative. Now, all of those children are born on February 29th. Now we can ask whether their mothers were also born on Leap Day, and we should expect that one in 1461 of them will have been, on average. So we should expect this two-million-to-one coincidence to occur around 130 times today. And remember, I made some quite conservative assumptions along the way. If on any given Leap Day it didn't happen that a child was born whose mother was also born on Leap Day, it would just be weird. In fact, once every eleven Leap Days or so we should expect a child born to parents both of whom were born on Leap Day.

There's another way to spot the unimpressiveness of this coincidence. For any given individual it's a two-million-to-one long shot that both they and their mother will have been born on Leap Day. But there are seven billion people in the world. So we would expect to find over 3000 such individuals alive at this moment. The world's a very big place, and something that's a two-million-to-one long shot can easily have enough opportunities to happen to be quite common overall.