The Strange Math That Predicts (Almost) Anything

The Strange Math That Predicts (Almost) Anything
Quizzed ESL Activities

Introduction to Probability and Markov Chains

A math feud in Russia over a hundred years ago sparked a division among mathematicians, with Pavl Necrosov, known as the “tsar of probability,” arguing that math could explain free will. This debate ultimately contributed to answers to various complex questions, including shuffling cards, building nuclear bombs, and predicting search results.

The Law of Large Numbers

The law of large numbers, which states that the average outcome of independent trials gets closer to the expected value as the number of trials increases, was a key concept in this debate. Jacob Bernoulli’s concept in 1713 proved that the law of large numbers worked for independent events. However, when events become dependent, the average no longer converges to the true value.

Necrosov and Marov’s Feud

Necrosov and Andre Marov, an atheist, had a feud over the concept of free will and the will of God. Marov criticized Necrosov’s use of mathematics to explain these ideas, considering it an abuse of math. Their debate centered on the law of large numbers and the assumption of independence. Necrosov argued that seeing the law of large numbers indicates underlying independent events, while Marov disagreed and set out to prove that dependent events can also follow the law of large numbers.

Markov Chains

Marov analyzed the dependency of events in text by studying the poem “Eugene Anigan” by Alexander Pushkin. He found that the probability of certain letter combinations differed significantly from what would be expected if the letters were independent, thus demonstrating that the letters were dependent on each other. Marov created a dependent system, a chain of events, which still followed the law of large numbers, showing that convergence in social statistics doesn’t prove independent decisions or free will.

Applications of Markov Chains

Markov chains have been used to model various complex systems, including the spread of diseases and the behavior of particles. The concept of Markov chains was introduced by Andrey Markov and later played a significant role in major developments, including the Manhattan Project. Stanislav Ulam, a mathematician, used Markov chains to understand neutron behavior in nuclear bombs, which led to the development of the Monte Carlo method.

The Monte Carlo Method

The Monte Carlo method is a statistical method that approximates differential equations by generating random outcomes. It was developed by Ulam and Von Neumann and was highly successful in studying nuclear reactor designs. The method has since been used in various fields, including computer science and engineering.

Search Engines and PageRank

The founders of Google, Larry Page and Sergey Brin, used Markov chains to develop the PageRank algorithm, which ranks web pages by relevance and quality. The algorithm models the web as a Markov chain, where links between pages represent transitions with certain probabilities. The PageRank algorithm has significantly improved search results and has become a crucial component of Google’s search engine.

Language Models and Predictive Text

Language models make predictions based on tokens, including letters, words, and punctuation, and use attention to focus on relevant context. These models have been used in various applications, including predictive text and language translation. However, they are vulnerable to repetitive processes and feedback loops, making them hard to model using Markov chains.

Conclusion

Markov chains are powerful tools for simplifying complex systems and making predictions due to their memoryless property. The concept of Markov chains originated from a mathematical dispute and has led to questions about randomness, such as how many shuffles it takes to randomize a deck of cards. Understanding the mathematics behind simple questions can lead to surprisingly complex concepts, which is what the platform Brilliant is all about.

Key Takeaways

  • The law of large numbers states that the average outcome of independent trials gets closer to the expected value as the number of trials increases.
  • Markov chains can be used to model dependent events and make predictions.
  • The Monte Carlo method is a statistical method that approximates differential equations by generating random outcomes.
  • PageRank is a algorithm that ranks web pages by relevance and quality using Markov chains.
  • Language models make predictions based on tokens and use attention to focus on relevant context.

Further Reading

For those interested in learning more about math, physics, programming, and AI, the platform Brilliant offers interactive lessons and challenges. A free 30-day trial and a 20% discount on their annual premium subscription are available.

Key Vocabulary

Term Definition Example Usage
Law of Large Numbers A principle stating that the average outcome of independent trials gets closer to the expected value as the number of trials increases. The law of large numbers can be observed in coin tossing, where the average outcome approaches 50% heads and 50% tails as the number of tosses increases.
Markov Chains A mathematical system that undergoes transitions from one state to another, where the probability of transitioning from one state to another is dependent on the current state. Markov chains can be used to model the behavior of particles in a nuclear reactor, where the probability of a particle transitioning from one state to another is dependent on its current state.
Monte Carlo Method A statistical method that approximates differential equations by generating random outcomes. The Monte Carlo method can be used to estimate the value of pi by generating random points within a square and counting the proportion of points that fall within a circle inscribed within the square.
PageRank An algorithm that ranks web pages by relevance and quality using Markov chains. PageRank can be used to rank web pages in a search engine results page, where the ranking is based on the probability of a user transitioning from one page to another.
Language Models A type of artificial intelligence model that makes predictions based on tokens, including letters, words, and punctuation, and uses attention to focus on relevant context. Language models can be used in predictive text applications, where the model predicts the next word in a sentence based on the context of the previous words.
Independent Events Events that do not affect the probability of each other occurring. Coin tossing and rolling a die are examples of independent events, where the outcome of one event does not affect the outcome of the other.
Dependent Events Events that affect the probability of each other occurring. Drawing cards from a deck without replacement is an example of dependent events, where the probability of drawing a certain card is affected by the previous cards drawn.
Expected Value The average value of a random variable over a large number of trials. The expected value of rolling a fair die is 3.5, which is the average value of the possible outcomes (1, 2, 3, 4, 5, and 6) over a large number of rolls.
Convergence The process of a sequence of values approaching a limit as the number of trials increases. The law of large numbers states that the average outcome of independent trials converges to the expected value as the number of trials increases.

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The Strange Math That Predicts (Almost) Anything

Vocabulary Quiz

1. What does the law of large numbers state about independent trials?

A) The average outcome gets farther from the expected value as trials increase
B) The average outcome remains constant regardless of the number of trials
C) The average outcome gets closer to the expected value as the number of trials increases
D) The average outcome is always equal to the expected value

2. Who developed the concept of Markov chains and applied it to the study of text dependency?

A) Pavl Necrosov
B) Jacob Bernoulli
C) Andre Marov
D) Andrey Markov

3. What statistical method was developed by Stanislav Ulam and Von Neumann to approximate differential equations?

A) The Law of Large Numbers Method
B) The Markov Chain Method
C) The Monte Carlo Method
D) The PageRank Algorithm

4. What algorithm, developed by Larry Page and Sergey Brin, uses Markov chains to rank web pages by relevance and quality?

A) The Monte Carlo Algorithm
B) The Markov Chain Algorithm
C) The PageRank Algorithm
D) The Language Model Algorithm

5. What is a key property of Markov chains that makes them useful for simplifying complex systems?

A) Memory retention
B) Dependence on initial conditions
C) Memoryless property
D) Sensitivity to feedback loops

Answer Key:

1. C
2. D
3. C
4. C
5. C

Grammar Focus

Grammar Focus: The Use of the Present Perfect Tense to Describe Completed Actions with a Connection to the Present

The present perfect tense is used to describe actions that started in the past and have a connection to the present. It is formed using the present tense of the auxiliary verb “have” and the past participle of the main verb. For example, in the text, “Their debate centered on the law of large numbers and the assumption of independence” can be rephrased using the present perfect tense as “Their debate has centered on the law of large numbers and the assumption of independence,” implying that the debate started in the past and continues up to the present. Another example from the text is “Marov has created a dependent system, a chain of events,” which indicates that Marov created the system in the past and it still exists or has an impact in the present. The present perfect tense is often used with words like “since,” “for,” “just,” “already,” “yet,” and “ever” to provide more information about the timing of the action.

Grammar Quiz:

Choose the correct answer for each question:

  1. By the time Andrey Markov introduced the concept of Markov chains,
  2. The founders of Google and significantly improved search results.
  3. Stanislav Ulam and led to the development of the Monte Carlo method.
  4. Language models in various applications.
  5. The platform Brilliant

Answer Key:

  1. D) mathematicians had already developed similar concepts
  2. A) used Markov chains to develop the PageRank algorithm
  3. A) used Markov chains to understand neutron behavior in nuclear bombs
  4. A) make predictions based on tokens and use attention to focus on relevant context
  5. C) is offering interactive lessons and challenges for those interested in learning more about math, physics, programming, and AI