Falling in love by the end of this song - doing the math!
Mood: nerdy
Posted on 2013-10-26 16:19:00
Tags: math
Words: 544

But somewhere in the world someone is gonna fall in love by the end of this song,
--"Odds Are" by Barenaked Ladies

We saw the Barenaked Ladies in concert this week (it was fun!) and not for the first time I heard that line and wondered if that was true. So let's do the math!

For these purposes, I'm going to define "falling in love" as "something that leads to a wedding". [1] There are 7.12 billion people in the world[2], and average world life expectancy is 70 years.[3] The average number of marriages a person has throughout her life is a little tricky to track down, but I'll go with the estimate of 1.21.[4] Again to simplify things, I'm going to assume these marriages as evenly distributed throughout one's life.[5]

So, each person will have an average of 1.21/70 = 0.017286[6] falling in loves per year. Multiplying by 7.12 billion (and dividing by 2, assuming each "falling in love" involves exactly 2 people) gives us around 61.5 million falling in loves per year, or 168,500 falling in loves per day.

That sounded really high to me, but after some thought (and checking the numbers if we just look at US couples), it looks about right. The world is a big place!

The song "Odds Are" is 182 seconds long[7], so on average if you start playing the song there will be (182 (seconds/song)/86400 (seconds/day))*(168,500 (falling in loves/day)) = 354.9 (falling in loves/song).[8]

Now while things are looking good for BNL's assertion, this just tells us the average, not the probability that this will happen. For that we need to look at the Poisson distribution. Here our gamma is 354.9, and the probability that we will get no events is

(354.9)^0*(e^-354.9)/(0!) = e^-354.9

Wolfram Alpha says that this probability is around 7*10^-155, so the odds that someone will fall in love by the end of this song are extremely good. (for some perspective, there are "only" around 10^80 atoms in the universe[9], so the chances of not having anyone fall in love during the song are about the same as if you pick two random atoms in the universe, and I do the same and happen to pick the same two atoms)

So: BNL is correct! Kudos.


References:

[1] - Yes, I realize you could define "falling in love" a bunch of different ways, but this one is one of the easier to count. Feel free to do your own math!
[2] - World Population article on Wikipedia
[3] - Quote by Colin Mathers, coordinator for mortality and burden of disease at the World Health Organization
[4] - naderalmaleh on Yahoo Answers. I would not recommend using this citation should you be writing a research paper!
[5] - Obviously this isn't true on an individual basis, but in aggregate it's not bad. To be more precise you'd have to come up with a reasonable distribution for a person and then look at how demographics are changing over time. If you want that kind of detail, I'd recommend sending this in to xkcd's what if? and getting off my back!
[6] - Math!
[7] - Odds Are article on Wikipedia
[8] - Hooray for dimensional analysis!
[9] - Wolfram Alpha query "atoms in the universe", because I got lazy.

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