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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 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.

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.

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.