By: Ali Binazir
June 15th, 2011

A little while ago I had the privilege of attending TEDx San Francisco, organized by the incomparable Christine Mason McCaull.  One of the talks was by Mel Robbins, a riotously funny self-help author and life coach with a syndicated radio show.  In it, she mentioned that scientists calculate the probability of your existing as you, today, at about one in 400 trillion (4×10¹⁴).

“That’s a pretty big number,” I thought to myself.  If I had 400 trillion pennies to my name, I could probably retire.

Previously, I had heard the Buddhist version of the probability of ‘this precious incarnation’.  Imagine there was one life preserver thrown somewhere in some ocean and there is exactly one turtle in all of these oceans, swimming underwater somewhere.  The probability that you came about and exist today is the same as that turtle sticking its head out of the water — in the middle of that life preserver.  On one try.

So I got curious: are either of these numbers correct?  Which one’s bigger?  Are they gross exaggerations?  Or is it possible that they underestimate the true number?

First, let us figure out the probability of one turtle sticking its head out of the one life preserver we toss out somewhere in the ocean.  That’s a pretty straightforward calculation.

According to WolframAlpha, the total area of oceans in the world is 3.409×10⁸square kilometers, or 340,900,000 km² (131.6 million square miles, for those benighted souls who still cling to user-hostile British measures).  Let’s say a life preserver’s hole is about 80cm in diameter, which would make the area inside

3.14(0.4)²=0.5024 m²

which we will conveniently round to 0.5 square meters.  If one square kilometer is a million square meters, then the probability of Mr Turtle sticking his head out of that life preserver is simply the area inside the life preserver divided by the total area of all oceans, or

0.5m²/3.409×10⁸x106m² = 1.47 x 10^-15

or one in 6.82×10¹⁴, or about 1 in 700 trillion.

One in 400 trillion vs one in 700 trillion?  I gotta say, the two numbers are pretty darn close, for such a farfetched notion from two completely different sources: old-time Buddhist scholars and present-day scientists.  They agree to within a factor of two!

So to the second question: how accurate is this number?  What would we come up with ourselves starting with first principles, making some reasonable assumptions and putting them all together?  That is, instead of making one big hand-waving gesture and pronouncing, “The answer is five hundred bazillion squintillion,” we make a series of sequentially-reasoned, smaller hand-waving gestures so as to make it all seem scientific. (This is also known as ‘consulting’ – especially if you show it all in a PowerPoint deck.)

Oh, this is going to be fun.

First, let’s talk about the probability of your parents meeting.  If they met one new person of the opposite sex every day from age 15 to 40, that would be about 10,000 people.  Let’s confine the pool of possible people they could meet to 1/10 of the world’s population twenty years go (one tenth of 4 billion = 400 million) so it considers not just the population of the US but that of the places they could have visited.  Half of those people, or 200 million, will be of the opposite sex.  So let’s say the probability of your parents meeting, ever, is 10,000 divided by 200 million:

10⁴/2×10⁸= 2×10^-4, or one in 20,000.

Probability of boy meeting girl: 1 in 20,000.

So far, so unlikely.

Now let’s say the chances of them actually talking to one another is one in 10.  And the chances of that turning into another meeting is about one in 10 also.  And the chances of that turning into a long-term relationship is also one in 10.  And the chances of that lasting long enough to result in offspring is one in 2.  So the probability of your parents’ chance meeting resulting in kids is about 1 in 2000.

Probability of same boy knocking up same girl: 1 in 2000.

So the combined probability is already around 1 in 40 million — long but not insurmountable odds.  Now things start getting interesting.  Why?  Because we’re about to deal with eggs and sperm, which come in large numbers.

Each sperm and each egg is genetically unique because of the process of meiosis; you are the result of the fusion of one particular egg with one particular sperm.  A fertile woman has 100,000 viable eggs on average.  A man will produce about 12 trillion sperm over the course of his reproductive lifetime.  Let’s say a third of those (4 trillion) are relevant to our calculation, since the sperm created after your mom hits menopause don’t count.  So the probability of that one sperm with half your name on it hitting that one egg with the other half of your name on it is

1/(100,000)(4 trillion)= 1/(10⁵)(4×10¹²)= 1 in 4 x 10¹⁷, or one in 400 quadrillion.

Probability of right sperm meeting right egg: 1 in 400 quadrillion.

But we’re just getting started.

Because the existence of you here now on planet earth presupposes another supremely unlikely and utterly undeniable chain of events.  Namely, that every one of your ancestors lived to reproductive age – going all the way back not just to the first Homo sapiens, first Homo erectus and Homo habilis, but all the way back to the first single-celled organism.  You are a representative of an unbroken lineage of life going back 4 billion years.

Let’s not get carried away here; we’ll just deal with the human lineage.  Say humans or humanoids have been around for about 3 million years, and that a generation is about 20 years.  That’s 150,000 generations.  Say that over the course of all human existence, the likelihood of any one human offspring to survive childhood and live to reproductive age and have at least one kid is 50:50 – 1 in 2.  Then what would be the chance of your particular lineage to have remained unbroken for 150,000 generations?

Well then, that would be one in 2^150,000 , which is about 1 in 10^45,000– a number so staggeringly large that my head hurts just writing it down. That number is not just larger than all of the particles in the universe – it’s larger than all the particles in the universe if each particle were itself a universe.

Probability of every one of your ancestors reproducing successfully: 1 in 10^45,000

But let’s think about this some more.  Remember the sperm-meeting-egg argument for the creation of you, since each gamete is unique?  Well, the right sperm also had to meet the right egg to create your grandparents.  Otherwise they’d be different people, and so would their children, who would then have had children who were similar to you but not quite you.  This is also true of your grandparents’ parents, and their grandparents, and so on till the beginning of time.  If even once the wrong sperm met the wrong egg, you would not be sitting here noodling online reading fascinating articles like this one.  It would be your cousin Jethro, and you never really liked him anyway.

That means in every step of your lineage, the probability of the right sperm meeting the right egg such that the exact right ancestor would be created that would end up creating you is one in 1200 trillion, which we’ll round down to 1000 trillion, or one quadrillion.

So now we must account for that for 150,000 generations by raising 400 quadrillion to the 150,000^th power:

[4x10¹⁷]^150,000 ≈ 10^2,640,000

That’s a ten followed by 2,640,000 zeroes, which would fill 11 volumes of a book the size of The Tao of Dating with zeroes.

To get the final answer, technically we need to multiply that by the 10^45,000 , 2000 and 20,000 up there, but those numbers are so shrimpy in comparison that it almost doesn’t matter.  For the sake of completeness:

(10^2,640,000)(10^45,000)(2000)(20,000) = 4x 10^2,685,007 ≈ 10^2,685,000

Probability of your existing at all: 1 in 10^2,685,000

As a comparison, the number of atoms in the body of an average male (80kg, 175 lb) is 10²⁷.  The number of atoms making up the earth is about 10⁵⁰.  The number of atoms in the known universe is estimated at 10⁸⁰.

So what’s the probability of your existing?  It’s the probability of 2 million people getting together – about the population of San Diego – each to play a game of dice with trillion-sided dice. They each roll the dice, and they all come up the exact same number – say, 550,343,279,001.

A miracle is an event so unlikely as to be almost impossible.  By that definition, I’ve just shown that you are a miracle.

Now go forth and feel and act like the miracle that you are.