The Strange Math That Predicts (Almost) Anything
What this covers
Veritasium traces how a 19th-century Russian mathematical disagreement spawned tools that now power everything from nuclear simulations to Google search and text prediction. Andrey Markov and Nikolai Nekrasov clashed over whether the law of large numbers—the statistical principle that averages stabilize over time—requires independent events. Markov settled the matter by analyzing 20,000 letters from Pushkin's Eugene Onegin, showing that even though vowel-consonant pairs were heavily dependent on each other, they still converged to predictable ratios. His insight led to Markov chains, a mathematical framework modeling systems where the next state depends only on the current one, not the full history before it.
The video follows how this one-century-old proof became foundational to three modern domains. In nuclear physics, John von Neumann realized neutron behavior couldn't be simulated independently and built the Monte Carlo method—running random simulations through a Markov chain to approximate solutions too complex for equations. In web search, Larry Page and Sergey Brin modeled the internet as a Markov chain where surfing patterns reveal page importance, yielding PageRank and upending early search engines that ranked by keyword repetition alone. In language, the video shows how predicting the next letter or word in a sequence is itself a Markov chain prediction problem, and how modern large language models extend this with attention mechanisms to handle longer context. The presenters also cover surprising applications: why shuffling a deck seven times suffices for randomness, and why feedback loops—like AI training on AI-generated text—become hard to predict with Markov chains.
Markov chains—mathematical tools for modeling dependent events—emerged from a 1905 Russian political feud and went on to underpin modern applications from nuclear weapons design to Google's search algorithm and large language models.
- Markov proved dependent events could follow the law of large numbers, overturning 200 years of probability theory that required independence
- Markov chains enabled Monte Carlo simulation for neutron behavior in nuclear bombs and later powered PageRank's ranking algorithm
- Modern AI language models use Markov chain principles with attention mechanisms to predict text, making them foundational to contemporary technology
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Markov analyzed the first 20,000 letters of Pushkin's 'Eugene Onegin' and found vowel-vowel pairs occurred only 6% of the time, far below the 18% expected if letters were independent, proving the letters were dependent on one another.
“if the letters were independent, the probability of a vowel-vowel pair would just be the probability of a vowel twice, which is about 0.18... But when Markov actually counted, he found vowel-vowel pairs only show up 6% of the time, way less than if they were independent.”
Markov's demonstration meant that observing convergence in social statistics does not prove the underlying decisions were independent, so such statistics don't prove free will at all; independence isn't even necessary to do probability.
“"Thus, free will is not necessary to do probability." In fact, independence isn't even necessary to do probability.”
Ulam had a flash of insight: instead of solving the Solitaire problem analytically, he could play hundreds of games and count how many could be won, which would give a statistical approximation of the answer.
“But then Ulam had a flash of insight, what if I just play hundreds of games and count how many could be won? That would give him some sort of statistical approximation of the answer.”
Markov built a two-state predictive machine using transition probabilities (e.g., 13% chance vowel-to-vowel, computed by dividing 0.06 by 0.43) and showed that running it produced a ratio converging to 43% vowels and 57% consonants—the exact split he had counted by hand—proving dependent events still follow the law of large numbers.
“after a while, it converges to a steady value, 43% vowels and 57% consonants, the exact split Markov had counted by hand. So Markov had built a dependent system, a literal chain of events, and he showed that it still followed the law of large numbers”
An overhand-style shuffle (the slow stacking motion shown) requires over 2,000 shuffles to randomize a deck, vastly more than the seven needed for a riffle shuffle.
“It's actually over 2,000... seven riffles or it doesn't count.”
Brin and Page modeled the web as a Markov chain by treating each link to a page as an endorsement (vote), where the more links a page sends out, the less valuable each individual vote becomes—measuring page importance by the long-run fraction of time a random surfer spends on each page.
“Brin and Page realized that each link to a page can be thought of as an endorsement. And the more links a page sends out, the less valuable each vote becomes. So what they realized is that we can model the web as a Markov chain.”
Running the neutron Markov chain on the ENIAC computer hundreds of times and tallying the multiplication factor k gives a statistical distribution: if k<1 the reaction dies, k=1 is self-sustaining, and k>1 grows exponentially into a bomb—approximating differential equations too hard to solve analytically.
“If you find that in most cases, k is less than one, the reaction dies down. If it's equal to one, there's a self-sustaining chain reaction... And if k is larger than one, the reaction grows exponentially and you've got a bomb.”
Yahoo's keyword search was easy to trick because you could repeat keywords hundreds of times hidden as white text on a white background; it had a notion of relevance but no notion of quality—which pages were actually better.
“Yahoo's keyword search was easy to trick. To get your page ranked highly, you could just repeat keywords hundreds of times, hidden with white text on a white background.”
Global warming involves a positive feedback loop: rising CO2 raises temperature, warmer atmosphere holds more water vapor (itself a powerful greenhouse gas), which raises temperature further—making the system hard to predict and hard to model with Markov chains.
“as the temperature increases, the atmosphere can hold more water vapor, which is an incredibly powerful greenhouse gas. And with more water vapor, the temperature increases further... So you get this positive feedback loop, which makes it hard to predict what's going to happen next.”
When events are dependent rather than independent—such as when people shout out guesses in public rather than submitting them individually—the average does not converge to the true value but instead clusters around a higher amount because each person is influenced by previous guesses.
“But now imagine that instead of asking each person to submit their guess individually, you ask people to shout out their answer in public. Well, in this case, the first person might think it's an extraordinarily valuable item, and say it's worth around $2,000, but now all the other people in the room are influenced by this value, and so, their guesses have become dependent. And now the average doesn't converge to the true value, but instead it clusters around a higher amount.”
The library endorsement model provides an analogy for measuring webpage quality: books with many date stamps (showing they were checked out frequently) acted as endorsements that the book was good, and the same idea can be applied to the web where each link to a page can be thought of as an endorsement.
“So I'm old enough that library books used to have a paper card in it that was a stamp of all the due dates of when it was due back. You took a book and if it had a lot of those, you said, 'Oh, this is probably a good book.' And if it didn't have any, you said, 'Well, maybe this isn't the best book.' Stamps acted like endorsements. The more stamps, the better the book must be. And the same idea can be applied to the web. Over at Stanford, two PhD students, Sergey Brin and Larry Page, were working on this exact problem. Brin and Page realized that each link to a page can be thought of as an endorsement.”
Markov's development of Markov chains should have been a huge breakthrough because in the real world almost everything is dependent on something else, including weather patterns, disease spread, and particle behavior, all of which can be modeled using Markov chains.
“This should have been a huge breakthrough, because in the real world, almost everything is dependent on something else. I mean, the weather tomorrow depends on the conditions today. How a disease spreads depends on who's infected right now, and the behavior of particles depends on the behavior of particles around them. Many of these processes could be modeled using Markov chains.”
Von Neumann realized that because a neutron's behavior depends on where it is and what it has done before, you cannot sample independent random outcomes like Solitaire; you needed a Markov chain modeling a whole sequence of events where each step influences the next.
“A neutron's behavior depends on where it is and what it has done before. So you couldn't just sample random outcomes like in Solitaire... What von Neumann realized is that you needed a Markov chain.”
Solitaire with all 52 cards has 52 factorial (~8×10^67) possible games, making the probability of winning impossible to solve analytically; Ulam's insight was to play hundreds of games and count winnable ones to get a statistical approximation.
“the total number of possible games was 52 factorial, or about eight times 10 to 67. So solving this analytically was hopeless. But then Ulam had a flash of insight, what if I just play hundreds of games and count how many could be won?”
The memoryless property—that you can ignore a system's entire long history and look only at the current state to predict the next—is what makes Markov chains powerful, because it lets extremely complex systems be radically simplified while still yielding meaningful predictions.
“You can just look at the current state and forget about the rest, that makes these systems memoryless. And it's this memoryless property that makes Markov chains so powerful because it's what allows you to take these extremely complex systems and simplify them a lot to still make meaningful predictions.”
For a deck of 52 cards, riffle shuffling it seven times makes every arrangement of the deck about equally likely, so it is effectively random; modeling card shuffling as a Markov chain (each arrangement a state, each shuffle a step) yields this result.
“for a deck of 52 cards, if you riffle shuffle it seven times, then every arrangement of the deck is about equally likely, so it's basically random.”
Claude Shannon found that predicting text improves as you condition on more preceding units: using two prior letters yields partial words, while using whole words as predictors produces locally sensible four-word sequences—demonstrating that higher-order Markov models make better text predictions.
“Shannon learned that you can make better and better predictions about what the next word is going to be by taking into account more and more of the previous words.”
Creating 100 pages all linking to your own page won't beat PageRank, because although those links briefly make your page seem important, no other quality websites link to those spam pages, so over many steps their contribution to the random surfer's time distribution becomes negligible.
“none of the other websites link to them. So over many steps, their contributions don't matter. You might have many links, but they're not quality links, so they don't affect the algorithm.”
Modern large language models operate on tokens (letters, words, or punctuation) rather than just an alphabet, predicting the odds of the next token given a string of prior tokens—but unlike simple Markov chains they use 'attention' to decide which prior tokens matter, allowing context like 'blood' and 'mitochondria' to disambiguate that 'cell' means biology rather than a prison.
“unlike simple Markov chains, they also use something called attention, which tells the model what to pay attention to... the model can use previous context like blood and mitochondria to know the cell most likely refers to biology rather than a prison cell.”
To avoid spam, simply creating 100 pages all linking to your website won't boost your ranking because while those new pages might initially seem important in the first few steps, none of the other websites link to them, so over many steps their contributions don't matter—you have many links but they're not quality links.
“It might seem like there's an easy way to beat the system, just make 100 pages all linking to your website. Now you get 100 full votes and you'll always rank on top, but that is not the case. While during their first few steps, they might make your page seem important, none of the other websites link to them. So over many steps, their contributions don't matter. You might have many links, but they're not quality links, so they don't affect the algorithm.”
Sergey Brin and Larry Page realized that not all links are equal: the more links a page sends out, the less valuable each vote becomes, because a link from a page that links to everything is worth less than a link from a page that links to few things.
“And the more links a page sends out, the less valuable each vote becomes. So what they realized is that we can model the web as a Markov chain.”
Modern large language models don't treat all tokens equally like simple Markov chains do; they also use something called attention, which tells the model what to pay attention to, allowing them to use distant context like previous words 'blood' and 'mitochondria' to know that 'cell' in the phrase 'the structure of the cell' refers to biology rather than a prison.
“But today's large language models don't treat all those tokens equally, because unlike simple Markov chains, they also use something called attention, which tells the model what to pay attention to. So in the phrase, 'the structure of the cell,' the model can use previous context like blood and mitochondria to know the cell most likely refers to biology rather than a prison cell.”
When people shout out their guesses publicly rather than submitting them individually, the guesses become dependent (anchored by the first answer) and the average no longer converges to the true value but clusters around a biased amount.
“all the other people in the room are influenced by this value, and so, their guesses have become dependent. And now the average doesn't converge to the true value, but instead it clusters around a higher amount.”
Bernoulli only proved the law of large numbers worked for independent events, where one event doesn't influence the others; for 200 years probability relied on the assumption that independence is required to observe the law of large numbers.
“But Bernoulli only proved that it worked for independent events like a fair coin flip... And so, for 200 years, probability had relied on this key assumption, that you need independence to observe the law of large numbers.”
When Google makes even the slightest change in its algorithms, it can have huge effects, demonstrating the critical importance of the Markov chain-based PageRank at the heart of Google's trillion-dollar business.
“When Google makes even the slightest change in its algorithms, it can have huge effects... And at the heart of this trillion dollar algorithm is a Markov chain, which only looks at the current state to predict what's going to happen next.”
Masayoshi Son realized that none of the leading search engines of the time had any superior technology and they all just ranked pages by how often a search term appears on a page, meaning the battle for the number one search engine would be decided by who could attract the most users and who could spend the most on marketing.
“See, Son had realized something. None of the leading search engines at the time had any superior technology. They didn't have a technological advantage over the others. They all just ranked pages by how often a search term appears on a given page. So the battle for the number one search engine would be decided by who could attract the most users, who could spend the most on marketing.”
Some in the industry thought Google's approach of getting the right result immediately was bad for business because if you get the answer on the first search, the user leaves and you get only one chance to show ads, whereas if it takes three or four searches, you get three or four chances to show ads.
“Although, to some, this sounded like a terrible idea... Others said, 'Oh, well you're telling me you get a search that will get the right result on the first answer? Well, I don't want that because if it takes them three or four chances, searches to get the right answer, then I have three or four chances to show ads, and if you get 'em the answer right away, I'm just gonna lose them.'”
Nekrasov took Bernoulli's idea one step further, arguing that if you observe the law of large numbers in data, you can infer that the underlying events must be independent.
“He said, if you see the law of large numbers, you can infer that the underlying events must be independent.”
PageRank adds a damping factor where 85% of the time the random surfer follows a link and 15% of the time jumps to a random page, ensuring all parts of the web are explored without the surfer getting stuck in a loop in disconnected networks.
“we can set a rule that 85% of the time, our random server just follows a link like normal. But then for about 15% of the time, they just jump to a page at random. This damping factor makes sure that we explore all possible parts of the web without ever getting stuck.”
Global warming illustrates a feedback loop problem: as CO2 increases, Earth's temperature increases, but higher temperature allows the atmosphere to hold more water vapor (a powerful greenhouse gas), which increases temperature further, creating a positive feedback loop that makes future climate states hard to predict.
“Take global warming, for instance, as we increase the amount of carbon dioxide in the air, the average temperature of the Earth increases. But as the temperature increases, the atmosphere can hold more water vapor, which is an incredibly powerful greenhouse gas. And with more water vapor, the temperature increases further allowing for even more water vapor. So you get this positive feedback loop, which makes it hard to predict what's going to happen next.”
If LLM-generated text increasingly fills the internet and becomes training data for future models, the system enters a feedback loop that collapses to a dull, stable state repeating the same thing forever—a vulnerability of language models that, like other feedback systems, becomes hard to model with Markov chains.
“you come, in this case, to us, a very dull, stable state, it just says the same thing over and over and over again forever. The language models are vulnerable to this process.”
A concern with large language models is that text they produce ends up on the internet and becomes training data for future models, creating a feedback loop where AI-generated content trains future AIs, making these systems vulnerable to degradation and potentially leading to very dull, stable states where the language model says the same thing over and over.
“But as large language models become more widespread, one concern is that the text they produce ends up on the internet and that becomes training data for future models... When you start doing that, the game is very soon over. You come, in this case, to us, a very dull, stable state, it just says the same thing over and over and over again forever. The language models are vulnerable to this process.”
Markov chains enable probability with dependent events, which matters because in the real world almost everything is dependent on something else—weather, disease spread, and particle behavior all depend on prior or surrounding states.
“This should have been a huge breakthrough, because in the real world, almost everything is dependent on something else.”
When Ulam returned to work at Los Alamos and shared his insight with John von Neumann, von Neumann immediately recognized its power but also spotted a key problem: in Solitaire each game is independent, but neutrons aren't independent—a neutron's behavior depends on where it is and what it has done before.
“When Ulam returned to work, he had a sudden revelation. What if we could simulate these systems by generating lots of random outcomes like I did with Solitaire? He shared this idea with von Neumann, who immediately recognized its power, but also spotted a key problem. See, in Solitaire, each game is independent. How the cards are dealt in one game have no effect on the next, but neutrons aren't like that. A neutron's behavior depends on where it is and what it has done before.”
Yahoo's search had a notion of relevance (whether a document talks about the thing you're interested in) but lacked a notion of quality (which results are better), which was the critical missing piece needed to rank pages effectively.
“One thing they didn't have in those early days was a notion of quality of the result. So they had a notion of relevance saying, does this document talk about the thing that you're interested in? But there wasn't really a notion of which ones are better.”
Any system with a feedback loop becomes hard to model using Markov chains because the future state depends not just on the current state but on the history of how the system has evolved, violating the memoryless property.
“And any system like this where we have a feedback loop, will become hard to model using Markov chains.”
As one paper put it, 'Problem-solving is often a matter of cooking up an appropriate Markov chain,' indicating that Markov chains are a fundamental problem-solving framework across many domains.
“As one paper put it, 'Problem-solving is often a matter of cooking up an appropriate Markov chain.'”
Shannon learned that you can make better and better predictions about what the next word will be by taking into account more and more of the previous words, a principle now used in Gmail's autocomplete feature that predicts what you'll type next, based on Markov chain algorithms.
“So Shannon learned that you can make better and better predictions about what the next word is going to be by taking into account more and more of the previous words. It's kind of like what Gmail does when it predicts what you're going to type next. And this is no coincidence, the algorithms that make these predictions are based on Markov chains.”
Markov created a predictive model with two states (vowel and consonant) and transition probabilities: from vowel, 13% chance to vowel and 87% chance to consonant; from consonant, 67% chance to consonant and 33% chance to vowel, which when run repeatedly converges to 43% vowels and 57% consonants.
“So he created a prediction machine of sorts... He started by drawing two circles, one for a vowel and one for a consonant... Markov knew that if you pick a random starting point, there is a 43% chance that it'll be a vowel. He also knew that vowel-vowel pairs occur about 6% of the time. So to find the probability of going from a vowel to another vowel, he divided 0.06 by 0.43 to find a transition probability of about 13%. And since there is a 100% chance that another letter comes next, all the arrows going from the same state need to add up to one. So the chance of going to a consonant is one minus 0.13, or 87%.”
The game for modern language model text prediction is: given a string of tokens (perhaps 30 long), what are the odds that the next token is this one, or that one, or the other one.
“The game is simply, we have this string of tokens that, you know, might be 30 long, and we're asking what are the odds that the next token is this or this or this?”
Yahoo had a critical weakness: its keyword search was easy to trick because to get your page ranked highly, you could just repeat keywords hundreds of times hidden with white text on a white background.
“But Yahoo had a critical weakness. See, Yahoo's keyword search was easy to trick. To get your page ranked highly, you could just repeat keywords hundreds of times, hidden with white text on a white background.”
A problem with simple Markov chain web crawling is that not all pages are connected and a random surfer can get stuck in a loop, never reaching the rest of the web, so PageRank fixed this by setting a rule that 85% of the time the surfer follows a link normally but 15% of the time they jump to a random page.
“But there is still one problem, though, not all pages are connected. In networks like this one, a random server can get stuck in a loop, never reaching the rest of the web. So to fix this, we can set a rule that 85% of the time, our random server just follows a link like normal. But then for about 15% of the time, they just jump to a page at random. This damping factor makes sure that we explore all possible parts of the web without ever getting stuck.”
Page and Brin disagreed with the idea that better search was bad for business, arguing that if their product was far superior, people would flock to it, and they believed they had technology to do a better job of search that would be impactful.
“But Page and Brin disagreed. They were convinced that if their product was far superior, then people would flock to it... we found that we had technology to do a better job of search, and we realized how impactful having great search can be.”
Early search engines like Yahoo had no superior technology—they all ranked pages merely by how often a search term appeared on a page—so the battle for the top search engine would be decided by marketing spend and user acquisition rather than technical merit.
“None of the leading search engines at the time had any superior technology... They all just ranked pages by how often a search term appears on a given page. So the battle for the number one search engine would be decided by who could attract the most users, who could spend the most on marketing.”
Some critics argued better search was bad for business because if users find the right result on the first search, the engine loses the three or four additional searches (and ad impressions) it would have shown otherwise.
“if it takes them three or four chances, searches to get the right answer, then I have three or four chances to show ads, and if you get 'em the answer right away, I'm just gonna lose them. So... I don't see why better search is better.”
Page argued PageRank is 'a democracy that works' because treating all pages as equal would let anyone manufacture billions of pages to game ranking, whereas weighting by quality links prevents manipulation.
“I would say it actually is a democracy that works. If all pages were equal, anybody can manufacture as many pages as they want. I can set up a billion pages in my server tomorrow. We shouldn't treat them all as equal.”
The law of large numbers states that as you run more and more independent trials, the average outcome gets closer and closer to the expected value.
“This behavior that the average outcome gets closer and closer to the expected value as you run more and more independent trials is known as the law of large numbers.”
A nuclear chain reaction occurs when a neutron splits a U-235 nucleus, releasing two or three more neutrons; if on average those neutrons go on to split more than one other U-235 nucleus, you get a runaway chain reaction and a nuclear bomb.
“If, on average, those new neutrons go on to hit and split more than one other U-235 nucleus, you get a runaway chain reaction, so you have a nuclear bomb.”
In the Russian poem 'Eugene Onegin' by Alexander Pushkin, when Markov analyzed the first 20,000 letters, he found 43% were vowels and 57% were consonants; when he examined vowel-vowel pairs, he found they occurred only 6% of the time, far less than the 18% expected if letters were independent.
“He took the first 20,000 letters of the poem, stripped out all punctuation and spaces, and pushed them together into one long string of characters. He counted the letters and found that 43% were vowels and 57% were consonants. Then Markov broke the string into overlapping pairs... Now, if the letters were independent, the probability of a vowel-vowel pair would just be the probability of a vowel twice, which is about 0.18 or an 18% chance. But when Markov actually counted, he found vowel-vowel pairs only show up 6% of the time, way less than if they were independent.”
The memoryless property of Markov chains—where you can ignore almost all the history of a system and just look at the current state to make meaningful predictions—is what makes Markov chains so powerful because it allows you to take extremely complex systems and simplify them greatly while still making meaningful predictions.
“But the beautiful thing Markov and others found is that for many of these systems you can ignore almost all of that. You can just look at the current state and forget about the rest, that makes these systems memoryless. And it's this memoryless property that makes Markov chains so powerful because it's what allows you to take these extremely complex systems and simplify them a lot to still make meaningful predictions.”
When shuffling with a less rigorous overhand shuffle (not a riffle shuffle), the number of shuffles needed is much higher, over 2,000, making it practically impossible to randomize by hand.
“How many times do you think you have to shuffle like this to get it random?... It's actually over 2,000.”
Nekrasov argued that if observed social statistics like marriage rates, crime rates, and birth rates follow the law of large numbers, this proves the underlying decisions must be independent, and therefore must be acts of free will—making free will measurable and scientific rather than merely philosophical.
“He said, if you see the law of large numbers, you can infer that the underlying events must be independent... Nekrasov reasoned that because these statistics followed the law of large numbers, the decisions causing them must be independent. In other words, he argued that they must be acts of free will. So to him, free will wasn't just something philosophical, it was something you could measure. It was scientific.”
The fact that Markov chains emerged from a mathematical feud in 1905 Russia that had nothing to do with their practical applications is remarkable; Markov's determination to show up Nekrasov led him to develop mathematics that would only decades later transform nuclear weapons design, search engines, and AI.
“It's kind of ridiculous to me that this basic fact of mathematics would come out of a fight like that, which, you know, really had nothing to do with it. But all the evidence suggests that it really was this determination to show up Nekrasov that led Markov to do the work.”
Nekrasov argued that because social statistics like marriages, crime rates, and birth rates follow the law of large numbers, the decisions causing them must be independent—and therefore acts of free will, making free will scientifically measurable.
“So Nekrasov reasoned that because these statistics followed the law of large numbers, the decisions causing them must be independent. In other words, he argued that they must be acts of free will.”
Understanding why 7 riffle shuffles randomize a deck requires understanding Markov chains—the interesting part isn't just knowing the answer but understanding why and seeing how a simple question can lead to surprisingly complex mathematics.
“But the interesting part isn't just knowing that, it's understanding why and seeing how a simple question can lead you to some surprisingly complex mathematics.”
The Monte Carlo method, named after the Monaco casino due to its random sampling and high stakes, spread quickly after the war—used by 1948 at Argonne for nuclear reactor designs—becoming a foundational simulation technique.
“the random sampling and high stakes reminded Ulam of the Monte Carlo Casino in Monaco, and the name stuck. The Monte Carlo method was born.”
Markov himself wrote 'I'm concerned only with questions of pure analysis. I refer to the question of the applicability with indifference,' showing he didn't care much about practical applications of his work.
“Markov himself seemingly didn't care much about how it might be applied to practical events. He wrote, 'I'm concerned only with questions of pure analysis. I refer to the question of the applicability with indifference.'”
When Jerry Yang declined Son's $100 million offer saying 'We don't need that much,' Masayoshi Son responded 'Jerry, everyone needs $100 million,' and when they didn't immediately accept, Son asked 'Who are your biggest competitors?' and after learning they were Excite and Lycos, said 'If you don't let me invest in Yahoo, I will invest in one of them and I'll kill you.'
“Jerry Yang declines saying, 'We don't need that much,' but Son disagrees, 'Jerry, everyone needs $100 million.' Before the founders get a chance to respond, Son jumps in again and asks, 'Who are your biggest competitors?' 'Excite and lycos,' the pair respond. Son orders his associate to write those names down. And then he says, 'If you don't let me invest in Yahoo, I will invest in one of them and I'll kill you.'”
In 1993, the internet was opened to the public, and by the mid-1990s thousands of new pages appeared every day in ever-expanding numbers, creating the problem of how to find anything in this sea of information.
“In 1993, the internet was open to the public, and soon it exploded. By the mid-1990s, thousands of new pages appeared every day, and that number was only growing. This created a new kind of problem. I mean, how do you find anything in this ever-expending sea of information?”
In 1994, Stanford PhD students Jerry Yang and David Filo founded Yahoo to solve the search problem, but when seeking $5 million in funding from Japanese billionaire Masayoshi Son (the 'Bill Gates of Japan'), Son instead offered $100 million—20 times more than requested.
“In 1994, two Stanford PhD students, Jerry Yang and David Filo, founded the search engine Yahoo, to address this issue, but they needed money. So a year later, they arranged to meet with Japanese billionaire, Masayoshi Son, also known as the Bill Gates of Japan. They were looking to raise $5 million for their next startup, but Son has other plans. He offers to invest a full $100 million instead. That's 20 times more than what the founders asked for.”
Within four years of accepting Son's investment, Yahoo became the most popular site on the planet, and the two founders became worth $120 million each.
“And within four years, Yahoo became the most popular site on the planet... the two men are now worth $120 million each.”
Andrey Markov, known as 'Andrey The Furious,' was an atheist mathematician who had no patience for people he considered unrigorous, and he publicly criticized Nekrasov's work, listing it among 'the abuses of mathematics,' arguing that math had nothing to do with free will or religion.
“His intellectual nemesis on the socialist side was Andrey Markov, also known as Andrey The Furious. Markov was an atheist and he had no patience for people who were being unrigorous, something he considered Nekrasov to be, because in his eyes, math had nothing to do with free will or religion. So he publicly criticized Nekrasov's work, listing it among 'the abuses of mathematics.'”
Uranium-235, the fissile fuel needed for bombs, was very hard to get, making a key question for nuclear weapon development how much uranium-235 was necessary to build a bomb, which is why Stanislaw Ulam wanted to understand how neutrons behave inside a nuclear core.
“But uranium-235, the fissile fuel needed for the bombs was really hard to get. So one of the key questions was just how much of it do you need to build a bomb? And this is why Ulam wanted to understand how the neutrons behave.”
On July 16, 1945, the United States detonated The Gadget, the world's first nuclear bomb, a six kilogram plutonium bomb that created an explosion equivalent to nearly 25,000 tons of TNT, as the culmination of the Manhattan Project.
“On the morning of the 16th of July, 1945, the United States detonated The Gadget, the world's first nuclear bomb. The six kilogram plutonium bomb created an explosion that was equivalent to nearly 25,000 tons of TNT. This was the culmination of the top secret Manhattan Project, a three-year long effort by some of the smartest people alive, including people like J. Robert Oppenheimer, John von Neumann, and a little known mathematician named Stanislaw Ulam.”
Pavel Nekrasov, the 'Tsar of Probability,' was a deeply religious and powerful mathematician who used his status to argue that mathematics could be used to explain free will and the will of God, making him the unofficial defender of the Tsar's position.
“On the side of the Tsar was Pavel Nekrasov, unofficially called the Tsar of Probability. Nekrasov was a deeply religious and powerful man, and he used his status to argue that math could be used to explain free will and the will of God.”
The Monte Carlo method was named after the Monte Carlo Casino in Monaco because Ulam's uncle was a gambler and the random sampling and high stakes reminded Ulam of the casino, and this name stuck.
“All that was needed was a name for the new method. Now, Ulam's uncle was a gambler, and the random sampling and high stakes reminded Ulam of the Monte Carlo Casino in Monaco, and the name stuck. The Monte Carlo method was born.”
Von Neumann and Ulam ran their Markov chain model on ENIAC, the world's first electronic computer, by randomly generating neutron starting conditions and stepping through the chain to track how many neutrons were produced on average per run (the multiplication factor k), then repeating this process hundreds of times and recording results in a histogram to get a statistical distribution.
“Next, they ran this chain on the world's first electronic computer, the ENIAC. The computer started by randomly generating a neutron starting conditions and stepped through the chain to keep track of how many neutrons were produced on average per run, known as the multiplication factor k... Then, after stepping through the full chain for a specified number of steps, we collect the average k-value and record that number in a histogram. This process was then repeated hundreds of times, and the results tallied up, giving you a statistical distribution of the outcome.”
In 1998, Page and Brin launched their search engine, initially calling it BackRub after the backlinks it analyzed, but then realizing that wasn't an attractive name, they chose Google from 'googol,' the largest number they could think of (10 to the power of 100), but accidentally misspelled it when registering their domain.
“And so, in 1998, they launched their new search engine to take on Yahoo. Initially, they called it BackRub, after the backlinks it analyzed, but then they realized that maybe that's not the most attractive name. Now, their ambitions were big to essentially index all the pages on the internet, and they needed a name equally as big. So they thought of the largest number they could think of, 10 to the power of 100, a googol. But then when trying to register their domain, they accidentally misspelled it. And so, Google was born.”
In 1905, socialist groups across Russia rose up against the Tsar, demanding complete political reform or his removal from power, which divided the nation into Tsarists defending the status quo and socialists seeking reform.
“In 1905, socialist groups all across Russia rose up against the Tsar, the ruler of the empire. They demanded a complete political reform, or failing that, that he stepped down from power entirely. This divided the nation into two.”
Google's name originated from 'googol' (10 to the power of 100), reflecting the founders' ambition to index all internet pages, but they accidentally misspelled it when registering the domain.
“they thought of the largest number they could think of, 10 to the power of 100, a googol. But then when trying to register their domain, they accidentally misspelled it. And so, Google was born.”
The Markov chain that powers Google's trillion-dollar ranking algorithm only looks at the current state to predict what happens next.
“at the heart of this trillion dollar algorithm is a Markov chain, which only looks at the current state to predict what's going to happen next.”
To randomize a standard deck of 52 cards using riffle shuffles, you need exactly 7 shuffles to make every arrangement about equally likely, making the deck essentially random.
“So you can think of card shuffling as a Markov chain where each deck arrangement is a state, and then each shuffle is a step. And so for a deck of 52 cards, if you riffle shuffle it seven times, then every arrangement of the deck is about equally likely, so it's basically random.”
A nuclear bomb works by having a core of uranium-235 where when a neutron hits a U-235 nucleus, the nucleus splits releasing energy and 2-3 more neutrons; if on average those new neutrons hit and split more than one other U-235 nucleus, you get a runaway chain reaction and a bomb.
“Now, a nuclear bomb works something like this. Say you have a core of uranium-235, then when a neutron hits a U-235 nucleus, the nucleus splits releasing energy and, crucially, two or three more neutrons. If, on average, those new neutrons go on to hit and split more than one other U-235 nucleus, you get a runaway chain reaction, so you have a nuclear bomb.”
Claude Shannon, the father of information theory, asked a different question in the 1940s: instead of using Markov chains on vowels and consonants, he focused on predicting individual letters, and when looking at only the last letter as a predictor, generated text that contained some recognizable words like 'whey,' 'of,' and 'the,' but mostly nonsense.
“But in the 1940s, Claude Shannon, the father of information theory, started asking a different question. He went back to Markov's original idea of predicting text, but instead of just using vowels and consonants, he focused on individual letters. And he wondered, what if instead of looking at only the last letter as a predictor, I look at the last two? Well, with that, he got text that looked like this. Now, it doesn't make much sense, but there are some recognizable words like 'whey', 'of', and 'the'.”
The Monte Carlo method was so successful that it didn't stay secret for long; by the end of 1948, scientists at Argonne Lab in Chicago used it to study nuclear reactor designs, and from there the idea spread quickly.
“The method was so successful that it didn't stay secret for long. By the end of 1948, scientists at another lab, Argonne, in Chicago, used it to study nuclear reactor designs, and from there, the idea spread quickly.”
For 200 years, probability theory had relied on the assumption that events must be independent to observe the law of large numbers, as proven by Jacob Bernoulli in 1713.
“This behavior that the average outcome gets closer and closer to the expected value as you run more and more independent trials is known as the law of large numbers. It was first proven by Jacob Bernoulli in 1713, and it was the key concept at the heart of probability theory right up until Markov and Nekrasov. But Bernoulli only proved that it worked for independent events like a fair coin flip.”
In January 1946, Stanislaw Ulam was struck by a sudden and severe case of encephalitis, an inflammation of the brain, that nearly killed him, resulting in a long and slow recovery during which he spent most of his time in bed.
“But then in January of 1946, everything came to a halt. Ulam was struck by a sudden and severe case of encephalitis, an inflammation of the brain, that nearly killed him. His recovery was long and slow, with Ulam spending most of his time in beds.”
The Solitaire problem Ulam faced was deceptively difficult because with 52 cards where each arrangement creates a unique game, the total number of possible games is 52 factorial, or approximately eight times 10 to the 67th power, making analytical solution hopeless.
“Ulam played with all 52 cards where each arrangement created a unique game, so the total number of possible games was 52 factorial, or about eight times 10 to 67. So solving this analytically was hopeless.”
In the Markov chain for neutrons, transition probabilities are not fixed but depend on things like the neutron's position, velocity, energy, and the overall configuration and mass of uranium; for example, a fast-moving neutron might have a 30% chance to scatter, 50% to be absorbed or leave, and 20% to cause fission, while a slower neutron would have different probabilities.
“But in this chain, the transition probabilities aren't fixed, they depend on things like the neutron's position, velocity and energy, as well as the overall configuration and mass of uranium. So a fast-moving neutron might have a 30% chance to scatter, a 50% chance to be absorbed or leave, and a 20% chance to cause fission. But a slower-moving neutron would have different probabilities.”
In a simple four-page web where Amy links only to Ben, Ben links to Amy, Chris, and Dan, and Chris and Dan both link to Ben, if you run the Markov chain and track time spent, Ben ranks first, followed by Amy, then Dan, and lastly Chris.
“To see how this works, imagine a toy internet with just four webpages. Call them Amy, Ben, Chris, and Dan... You spend the most time on Ben, so Ben is ranked first, followed by Amy, then Dan, and lastly Chris.”
Page and Brin built a better search engine called PageRank using Markov chains, which got much better search results, often getting you to the site you were looking for on the first try.
“By using Markov chains, Page and Brin had built a better search engine, and they called it PageRank... With PageRank, Google got much better search results, often getting you to the site you were looking for in one go.”
The name 'PageRank' serves a double purpose: it describes how pages react with each other (page rank as verb) and also includes founder Larry Page's name (page as noun), with Page sneaking his own name into the algorithm.
“Because it's talking about how pages react, webpages react with each other and also 'cause the founder's name is Larry Page, so he snuck that in.”
Over the next four years, Google overthrew Yahoo to become the most used search engine, and today Alphabet, Google's parent company, is worth around $2 trillion.
“Over the next four years, Google overthrew Yahoo to become the most used search engine... And today, Alphabet, which is Google's parent company, is worth around $2 trillion.”
Ulam later remarked: 'It is still an unending source of surprise for me to see how a few scribbles on a blackboard could change the course of human affairs,' reflecting on how his insight about Solitaire led to methods that transformed nuclear weapons and beyond.
“Ulam later remarked, 'It is still an unending source of surprise for me to see how a few scribbles on a blackboard could change the course of human affairs.'”