Estimating the Probability That an Explosion of Ink Prints a Dictionary

Braden Oh

A few years ago, I heard someone compare the probability of life originating by random chance to the probability of a dictionary being printed in the explosion of a printing shop. The numbers that person was using were clearly made up, but I still wondered – what are the odds? It turns out that quite a bit of scientific thought has been put towards calculating the probability of life arising [1-3] as well as calculating the probability of other highly unlikely events. These include the probability of The Flash quantum tunneling through a wall [4], a monkey with a typewriter producing the works of Shakespeare [5], and a fully intact brain appearing out of quantum fluctuations in the vacuum of space [6-7]. Yet somehow, no scientific thought has been given to the odds of an explosion in a print shop creating a dictionary. That seemed to me to be a great tragedy!

To construct a solid starting point for future researchers, I imagined a highly simplified abstraction of the print factory thought experiment: a sphere of ink explodes within a spherical shell composed of pieces of printing paper; the sphere contains exactly enough ink to print one copy of the 1989 Merriam-Webster English dictionary, and at the instant of the explosion, the sphere atomizes into droplets sized to print 300 dpi pixels.

The ink-sphere abstraction allows us to model the ink sphere as a sum of sectors.  Each sector is a rectangular pyramid that contains enough ink to print one page of the dictionary.  During the explosion a pyramid expands outwardly until its footprint perfectly covers a target piece of paper.  A diagram of this is shown in Figure 1.

Figure 1: A sphere of ink modeled as a sum of pyramidal sectors explodes inside a shell of printing paper

The sector abstraction allows us to model each pyramid as a stack of discrete sheets of ink droplets: the volume of a pyramidal sector is given by a 3D Riemann sum, and we can count the number of ink droplets directly!

The possible arrangements of these droplets are modeled by combinations and permutations that require the evaluation of enormous factorials to solve (e.g., 5,025,780!). These factorials are too difficult for a computer to evaluate by multiplication (even using tricks such as Stirling’s approximation) due to limitations in compute time, memory, and decimal precision.  I bypassed this problem by performing all of my computations in log space (since multiplication in linear space is addition in log space) with Python 3.8.5 floats, which have 17 decimal places of precision.  At the scale which emerged from these factorials my error was objectively enormous, yet it was still 15 orders of magnitude below 1%.

Using this model, I found the probability of a dictionary being printed to be 1.30×10^-817,692,555, or on the order of 10^{-10^9}.  More conventionally, this can be expressed as 1 in 7.68×10^817,692,554 (that is 7.68 followed by over 817 million zeroes)  Numbers at this order of magnitude are absurd to describe; for instance, 10^{10^9} is much, much larger than a googol, but much, much smaller than a googolplex.

To compare this probability to other unlikely events, the reported probability of The Flash quantum tunneling through a solid wall is approximately 10^{-10^28} [4], while the probability of a monkey randomly typing the complete works of Shakespeare is approximately 10^{-10^7.15} [6]. Hence, in a world where The Flash runs at a wall, a monkey has a typewriter, and spheres of ink are exploded, a dictionary will be printed well after the monkey types Shakespeare, but long before The Flash runs through the wall.  However, the universe will end many times over before any of these unlikely events occur.

I’ve shown my work in a full writeup available on the Frankly Speaking website.  I’ve also composed a technical paper which has been rejected by over half a dozen journals.  Do you have a suggestion for where I can send it next?  If not, I intend on publishing at SIGBOVIK 2023 this coming April.

[1] A. Loeb, et al. “Relative likelihood for life as a function of cosmic time.” 2016.

[2] D. Kipping. “An objective bayesian analysis of life’s early start and our late arrival.” 2020.

[3] C. Scharf, and L. Cronin. “Quantifying the origins of life on a planetary scale.” 2016.

[4] A. Alam, et al. “The flash and quantum tunnelling.” 2016.

[5] E. R. Weaver. “An exercise in probability.” 1965.

[6] C. Kittel and H. Kroemer. “Thermal Physics.” 1980.

[7] D. Page. “Return of the boltzmann brains.” 2006.

Full Paper:

Second installment (see the October issue online for Chapter 1).

Chapter 2

Within ten years the first civilians made the trip from Venus to Cyro; within twenty years twenty ships a day ferried passengers and goods; and within fifty years it was two hundred ships a day. In a hundred years, the ships were as big as container ships carrying kilotons of cargo from one planet where it was abundant to the other where it was scarce.

Contrary to the imagination of the first diplomats, the cargo did not consist primarily of containers of hot and cold air from the respective planets’ atmospheres. To be sure, there were forms of bulk heat transfer to power the refrigeration units at Venus’ chemical plants and parts of Cyro’s steel industry, but even with advanced heat transfer mechanisms, ferrying Kelvins via rocket was hardly economical. In the words of a would-be entrepreneur, “Schlepping gold just isn’t lucrative when the gold in question is lighter than air.”

However, the two planets had natural resources and industries that complemented each other quite perfectly in many ways, and the expedience of the new avenue of trade would have brought tears of delight to any economist of the period. Cyro churned out coal and iron ore and sent it to Venus where the furnaces were cheaper to operate. Venus had the sand to make glass but not the technology to make it strong and insulated like the marble windows Cyro’s cities were known for. Cyro had advanced research in pulmonology after an airborne disease had ravaged their population generations ago; Venusian teeth sparkled and seduced because Venus was home to acclaimed schools of dentistry. On and on the examples went, without limit, and increasing all the time.

Bi-globalization, as it was called, brought on another industrial revolution for both planets. Prosperity increased for everyone, but especially for the wealthy, and life expectancies rose across the board. New diseases naturally spread from one culture to the other after millenia without contact, but with them spread new cures, new drugs, and new vaccines, the subsidized sale of which generated enough money for the pharmaceuticals that they could parley the official number of cases down to about ten percent so that nobody knew exactly how much they were paying.

Around this time, a new currency was minted by Venus’ Planetary Government to mirror the system in place on Cyro: not so much a currency as a communal ledger, to replace the old custom of carrying around a tank of cold gas to pay for things. Every house in every city on Venus was by now connected to a central cold reservoir which was maintained by the government at great cost using the most sophisticated refrigeration system known to humankind. Everyone was allocated a share of the cold reservoir, according to their wealth, and anyone could make purchases or transfer their currency to another person by flipping a switch on their wall and receiving a blast of hot air into their living room as coldness was sucked into the system, or they could withdraw funds to bathe at their leisure in chilly air.

When they made the switch to the communal ledger, there was quite an uproar from certain posh parts of the capital city, because the government’s assessment of each person’s wealth—on which allocations of the new currency were made out—was based on information collected from the last year’s tax returns. It came to light that certain individuals’ actual wealth exceeded the numbers reflected on their tax forms by a factor of at least tenfold, and those individuals made a swine’s stink about being swindled by the government’s allocation scheme. The mayor pointed out that if they wanted the government to allocate the new currency correctly, perhaps they should not have deceived the government on their tax returns, and the certain individuals hired lawyers and went on a witch hunt as the press called it for other anomalies in accounting. They found a handful of isolated cases where other people, mostly lower class, had been paid too little, but overall the lawyers only made a fool of themselves, and even still they sued the government and won.

The transport ships were owned by a private company named Guildman, which had gotten its start two centuries ago selling cooling mechanisms for Venus’ wealthiest homes. The first commercial devices used water for evaporative cooling, which made them prohibitively expensive; later, around the same time as the printing press, they learned to use an energy source to drive a reusable fluid through a compression cycle to take heat out of the air. (History textbooks, whose authors never disclosed how well they were paid, declared that it was the Air Conditioner, not the Printing Press, that marked the end of the Middle Ages on Venus.)

Harrison Guildman, the founder and namesake of the largest monopoly on both planets, was three hundred years old and had been dead for two hundred and thirty, but that did not bother him; to this day, he steered the company with an iron grip. Since the founder’s untimely death, no new chairman had ever been elected. None needed to be. For in his last days, spent alone without food nor drink nor certainly sleep, he had written out a strategic plan for the company which was so detailed and forward-thinking that no one had needed to take his place at the helm for two whole centuries. The actual plan remained highly classified and had only ever been read in its entirety by a dozen or so of its top executives as they carried out his words. But certain passages had been made public for promotional events, including for the first time when Guildman announced that it would be dedicating its facilities to supporting the Ferry Commission in its bid for building the first interplanetary shuttle. 

It was right there in Harrison Guildman’s own handwriting, on parchment cracked and the ink of two hundred years ago faint but still legible: “Shall the government of Venus ever deem it of national importance to establish communication and trade with the White Planet through the means of space travel, the Guildman Corporation shall be the first to announce its full support for the program and dedicate at least one-half of its research facilities and budget to the mission.” As soon as the project was underway, and Guildman’s indispensability to the project had been established, it was revealed that the document in fact continued: “In exchange, the Guildman Corporation will demand full ownership of the space vehicles under development, although the ports themselves, being a matter of national security, may remain under government operation and control.” And no one had much choice but to go along.

To accommodate the boom in trade, the Venus-Cyro Transportation Authority (the first inter-planetary, bi-governmental body) worked frantically to expand its ports on both planets. Another super-port was built from the ground up every five years, along with a whole new city to match. At first they kept up with demand, but the ports’ infrastructure was built hastily and not to last, so they soon fell into debt and Guildman took over the operation of the ports, too.

It was never established whether this final consolidation had been prophesied in the founding documents, but cynics didn’t put it past the man who had thought of every possibility at once, while at the same time thinking of nothing at all but his company’s own insatiable expansion.