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Borel’s Law of Probability

I've spent half a century writing for radio and print (mostly print). I hope to still be tapping the keys as I take my last breath.

Borel’s Law is about the probability of events occurring.

Borel’s Law is about the probability of events occurring.

In 1943, the distinguished French mathematician Émile Borel developed a law about probabilities that stated “events with a sufficiently small probability never occur” (Institute of Mathematical Statistics). He used a thought experiment to illustrate this that became known popularly as the “infinite monkey theorem;” this states that if an infinite number of monkeys pound the keys of an infinite number of typewriters they will eventually write the complete works of Shakespeare.

Borel’s Law has since been enlisted by creationists and evolutionists alike to bolster their arguments.

Borel’s Law for Non-Mathematicians

Those who are brave (foolish?) enough to delve into higher mathematics discover there are many tripwires ahead of them. They look like this ∑, or this∮, and have to be avoided at all costs.

So, who better to explain probability theory than someone who is a complete duffer at mathematics? Fortunately, just such a person is poised at the keyboard right now, so let’s get started. If this writer can grasp the concept then any one of those infinite monkeys can.

Essentially, what Borel said was that any event with a honking big (a technical term used by mathematicians) level of improbability would never happen. The learned Frenchman put a number on it―10 to the power of 50, written as 1050―in order to impress upon the common herd that its members are not mathematicians.

For the curious, that is expressed as one in 100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000. Anything with a lower probability than that would not happen, said Borel the numbers man.

Creationists Use Borel’s Law

Those who say Charles Darwin’s concept of evolution is hogwash gleefully seize on Borel’s Law to support their arguments.

They say that it is impossible for human life to exist without divine intervention. The first single-cell organism emerging from an inanimate chemical soup is not something that could have happened by chance. As Borel pointed out such an event was so improbable as to be impossible.

Scott and Janice Huse, in their 1997 book The Collapse of Evolution, state that “It is very significant to note that mathematicians generally consider that any event with a probability of one chance [in] 10^50 as having a zero probability (i.e. it is impossible).”

The astronomer Sir Fred Hoyle illustrated this with his Junkyard Tornado Theory: “The chance that higher life forms might have emerged in this way is comparable to the chance that a tornado sweeping through a junkyard might assemble a Boeing 747 from the materials therein.”

Your Existence Is Impossible

If Borel’s Law is the immutable truth and the creationists are wrong, you can’t exist. However, as the astute who do exist will have observed, extremely improbable events do actually occur.

Has anybody ever said to you “You’re one in a million”? Me neither. But, despite your being an extremely wonderful person, such a statement is wildly inaccurate. One number that is tossed about a lot is that the odds against your being born are one in 400 trillion. But, doesn’t that seem a bit low? Dr. Ali Binazir, who describes himself as a happiness engineer, thinks it’s way off the mark.

In a 2011 HuffPost article, he set about calculating the likelihood of each of us being born. He wrote that a “supremely unlikely and utterly undeniable chain of events” had to take place before the sperm with half your name on it met up with the egg with the other half.

That chain involved every ancestor, all the way back to the original hominids, getting romantic at precisely the right moment to keep the sequence going that produced you. That’s three billion years, or about 150,000 generations, of reproduction without a hitch.

Dr. Binazir calculated that the odds against each of us being born produced a number that makes the brain hurt. So he gave us an analogy that helps: “It’s the probability of 2.5 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.” This is a vastly greater improbability than one in 1050.

Borel’s Law says such a number means something is impossible, and yet, it’s not. Because there you are futzing about on the internet reading incredibly interesting articles like this one.

"It is a part of probability that many improbable things will happen.”

— Aristotle

The Influence of Big Numbers

A rational approach acknowledges that incredibly low probabilities is not the same as zero probability.

The likelihood that improbable events occur is controlled by the scale of the Universe. It was always likely that a living cell would jump out of that primordial soup because the conditions for that to happen must have existed somewhere; and, probably, in several somewheres.

Our own galaxy, the Milky Way, has as many as 400 billion stars in it and at least 100 billion planets. Astronomers estimate that there are at least 100 billion galaxies in the observable Universe. That’s just the observable Universe; we haven’t the faintest idea what is beyond what we can detect with our instruments.

So, it seems fair to say that there are an infinite number of possibilities of any event happening not matter how remote the chance.

Here’s how the National Center for Science Education puts it: “Any event with a probability greater than 0, no matter how low, will be likely to happen if given enough opportunity, and sure to happen if opportunity is unlimited.”

Bonus Factoids

  • Mathematician Professor John Littlewood of Cambridge University defined a miracle as an event happening with a frequency of one in a million. He calculated that an average human could expect to experience such an occurrence once every 35 days. His reasoning being that each person experiences an event of some sort every second. He assumes each person is alert and awake for eight hours a day (this allows for downtime watching reality TV shows when the human brain is entirely inactive). So, that’s 28,800 events a day, adding up to a million in 35 days. The learned professor was actually pulling everyone’s legs, but Littlewood’s Law has been conscripted as “proof” of a number of strange theories.
  • The perfect deal in bridge is that each player receives all the cards in one suit. The probability of this happening is 635,013,559,600 to one against. But, the odds of every bridge deal are exactly the same.
  • Gamblers always play the odds; their lives revolve around probabilities, and that has led many into dark places. In 1913, at the roulette wheel at the Casino de Monte-Carlo, the ball dropped into a black slot 26 times in succession. Fortunes were lost as players bet huge amounts on red in the erroneous belief that the law of probabilities dictated the ball would not drop on black again. The odds against 26 blacks in a row are about 66 million to one against; however, previous results have absolutely no effect on subsequent ones. The odds of red or black are 50:50 with each spin of the wheel.


This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 2020 Rupert Taylor


Rich E Cunningham from Ontario, Canada on May 14, 2020:

Interesting article, guess we are lucky to be here!

Miebakagh Fiberesima from Port Harcourt, Rivers State, NIGERIA. on May 07, 2020:

Rupert, in spite of too many odds in a probability, bottom line is that it is just 2 numbers of the roulette, black and red. If the first shot fall into the black hole, it remains only the red, and not green or blue. Moreover, the second shot would not necessary run into the red. But on all probability it would. Here it should be realize it is a quest work. That is the reason the majority dislike games, lotto, and sports beting. As you aptly point out, many persons losts lots of money at the roulette because they quests mathematically accurate. Probability is a pure odd. It is like finding out how many angels can stand on a pin-head! A lovely day to you.