The hottest Probability Substack posts right now

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The Monty Hall problem: The golden goat variation

Simplicity is SOTA • 131 implied HN points • 03 Feb 25
🔬 Science Mathematics Probability Bayesian Logic Puzzles
  1. The Monty Hall problem has a new twist, focusing on a valuable goat instead of a car. In this version, knowing which goat is valuable affects your choice.
  2. Using Bayes' theorem can help calculate the probabilities in this variation. After a goat is revealed, you can reassess your chances to make a better decision.
  3. The essential lesson is to update your beliefs with new information. Recognizing how new clues impact your choices is key to making smarter decisions.

All Bayes Everything

Holodoxa • 239 implied HN points • 14 Jun 24
🔬 Science Statistics Probability Neurobiology Consciousness Research
  1. Bayes' Theorem is a powerful concept in probability theory that helps update beliefs based on new evidence, highlighting the importance of combining prior knowledge and new data.
  2. Bayesian methods can offer valuable improvements to scientific research practices by emphasizing uncertainty, effect magnitude, and probability distributions over traditional p-values and null hypothesis testing.
  3. The concept of the brain functioning as a prediction machine aligns with Bayesian principles, suggesting that the brain uses prior knowledge and new sensory inputs to make predictions and construct conscious experiences.

Chutes, ladders, and Markov chains

A Piece of the Pi: mathematics explained • 72 implied HN points • 04 Dec 24
🚌 Education Mathematics Statistics Game Theory Probability Teaching
  1. The game of Chutes and Ladders is a fun example of a Markov chain. It shows how the next move depends only on where you are now, not on how you got there.
  2. There are different types of game boards, some allow for winning while others can trap players forever. Ultimately winnable boards guarantee that a player can reach the end if they keep playing.
  3. On average, players need about 39 spins to win the game, and surprisingly, most random boards created will still offer a winning chance.

There are Complex-Number One-Norm Square-Root of Probability Amplitudes of 0.006 in Which Sam Bankman-Fried Is Happy

Brad DeLong's Grasping Reality • 676 implied HN points • 05 Oct 23
💰 Finance Investing Risk management Probability Psychology Trading
  1. Amplitudes in quantum-mechanical superposition relate to philosophy-of-probability vs. psychology.
  2. Understanding the Kelly Criterion for betting based on win-loss odds and maximizing returns.
  3. Traders use the Kelly Criterion for survival, making positive-value bets, and psychological factors.
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Law of Large Numbers and the Gamblers Fallacy[Math Mondays]

Technology Made Simple • 179 implied HN points • 11 Sep 23
🚌 Education Math Probability Theory Computer Science Technology
  1. The Law of Large Numbers states that as the number of trials increase, the average of results will get closer to the expected value.
  2. This law is crucial in scientific fields, allowing predictions on chaotic events, leading to industries like gambling and insurance.
  3. Misunderstanding the Law of Large Numbers can lead to the Gambler's Fallacy, as it deals with the convergence of infinitely many experiments, not individual ones.

Math Problems

The Better Letter • 157 implied HN points • 17 Mar 23
🔬 Science Mathematics Probability Sports Psychology Economics
  1. Unlikely events happen more often than we realize, influencing outcomes in sports, investments, and life.
  2. Probability plays a significant role in determining outcomes, such as in coin tosses, NCAA brackets, and market predictions.
  3. Randomness, noise, and unpredictability are intrinsic to life, affecting decision-making and the way we perceive events.

Yet Another Ball and Urn Problem

Metarational • 59 implied HN points • 13 Feb 24
🔬 Science Probability Mathematics Puzzle
  1. The problem involves repeatedly selecting balls from an urn, inspecting their color, putting them back, and adding another of the same color. The goal is to find the probability that the majority of balls in the urn will be white after a large number of repetitions.
  2. To solve the problem, it was analyzed that there must be at least half white draws to achieve a white majority. Calculations led to a final result of 11/16 as the probability limit.
  3. The solution involved understanding the probabilities of different color sequences and using Riemann sums to simplify and find the answer, showcasing an intricate application of mathematics to a probability riddle.

My Talk About the Sleeping Beauty Paradox

By Reason Alone • 16 implied HN points • 07 Nov 24
📖 Philosophy Probability Decision Theory Cosmology Ethics Rationality
  1. The Sleeping Beauty paradox involves a coin flip that affects how often she wakes up, which raises questions about probability. People have different opinions on how she should assess the chance of heads when she wakes up.
  2. One group, called 'halfers', believes the chance of heads remains 50/50 since she doesn't gain new information about the coin when waking up.
  3. Another group, 'thirders', argues she should think there's a one in three chance it's heads because of how many times she might wake up, depending on the coin flip.

Tanks for the memories

Logging the World • 179 implied HN points • 11 Dec 22
🔬 Science Statistics Probability Military History Cybersecurity Data Analysis
  1. In a raffle with a large number of tickets, the biggest number drawn out starts to show some structure as more tickets are selected.
  2. By looking at the maximum value drawn in a raffle, one can estimate the total number of tickets, a concept applied in statistics like the German tank problem.
  3. Sequential numbering schemes can reveal interesting insights, as seen in situations like the Skripal poisonings and Novak Djokovic's COVID test, highlighting the importance of careful numbering practices.

Log Odds or Probability

Mindful Modeler • 139 implied HN points • 25 Apr 23
🔬 Science Statistics Interpretation Regression Probability
  1. Log odds are additive, probabilities are multiplicative. Some interpretation methods like expressing predictions as a linear sum may benefit from log odds.
  2. Edge transitions, like from 0.001 to 0.01, may sometimes be more significant than middle transitions, like 0.5 to 0.6.
  3. Probabilities offer intuitive understanding for decision-making, cost calculations, and are more commonly familiar compared to log odds.

A model of models of probability

Bram’s Thoughts • 78 implied HN points • 23 Nov 23
🔬 Science Probability Models Human behavior
  1. People generally have a simplified internal model of probability with five main categories.
  2. People tend to struggle with accurately gauging differences in expected values within the 40-60% range.
  3. Individuals often display overconfidence in their predictions for probable events and can become overly upset when these predictions fail.

Understanding Different Uncertainty Mindsets

Mindful Modeler • 179 implied HN points • 24 Jan 23
🔬 Science Probability Modeling Uncertainty Machine Learning Statistics
  1. Understanding the fundamental difference between Bayesian and frequentist interpretations of probability is crucial for grasping uncertainty quantification techniques.
  2. Conformal prediction offers prediction regions with a frequentist interpretation, similar to confidence intervals in linear regression models.
  3. Conformal prediction shares similarities with the evaluation requirements and mindset of supervised machine learning, emphasizing the importance of separate calibration and ground truth data.

What is Bayesian Statistics? The Beginner Math Guide (Part One)

The Software & Data Spectrum • 78 implied HN points • 13 Apr 23
🔬 Science Statistics Probability Logic Combinatorics
  1. Bayesian Statistics is used in various fields like Machine Learning, Engineering, Data Science, and more.
  2. Bayesian Thinking involves observing data, holding prior beliefs, forming hypotheses, gathering evidence, and comparing hypotheses.
  3. Probability is a way to measure belief strength, and calculating probabilities involves counting outcomes and using ratios of beliefs.

How I once won the lottery

inexactscience • 19 implied HN points • 06 Sep 23
🚌 Education Mathematics Psychology Probability Statistics
  1. Sticking to one choice in a lottery doesn't change your odds, which stay at 1 in 24 no matter what. It seems like it should matter, but it really doesn't.
  2. If a lottery is unfair and avoids streaks, choosing the same number can actually be a better strategy because it decreases your risk of never winning.
  3. Many people fall for the gambler's fallacy, thinking just because a number hasn't won in a while, it should win soon. But in a fair lottery, each draw is independent and has the same odds.

The most important assumption in statistics- IID [Math Mondays]

Technology Made Simple • 39 implied HN points • 01 Aug 22
🔬 Science Statistics Probability Machine Learning
  1. The most important assumption in statistics is IID, which stands for Independently and Identically Distributed
  2. IID assumption is crucial for statistical analysis - it helps in making accurate deductions and avoiding mistakes, like the gambler's fallacy
  3. Understanding IID involves recognizing independent and identical distributions in data samples, which are essential for various statistical techniques

Siêu trí tuệ: tìm số nguyên tố hay là tăng minh bạch cho ngành đánh đề

Thái | Hacker | Kỹ sư tin tặc • 39 implied HN points • 27 Dec 19
🔬 Science Mathematics Algorithms Randomness Probability Cryptography
  1. When faced with challenges involving prime numbers, clever algorithms can help quickly eliminate composite numbers and pinpoint the secret numbers.
  2. The difficulty of a problem depends on the randomness of number selection within a matrix and the position of prime numbers.
  3. Designing a fair random number generation system is crucial for ensuring transparency, not only in intellectual competitions but also in traditional gambling industries.

Evidence Belongs on the Canvas Not the Scale

Metarational • 19 implied HN points • 20 Apr 21
🔬 Science Mathematics Probability Evidence Metaphor
  1. Evaluating evidence like weighing it on a balance scale can be an elegant metaphor but may not be mathematically correct, as evidence doesn't always work that way.
  2. The scenario with two judges deliberating on a statement showcases how evidence overlap matters, revealing flaws in the scale metaphor and emphasizing the need for a more nuanced model.
  3. Imagining evidence on a canvas with shaded regions for different hypotheses can better capture the complexity of multiple evidence lines overlapping, offering a more accurate representation than a simple scale.

How large that number in the Law of Large Numbers is?

The Palindrome • 4 implied HN points • 04 Sep 23
🔬 Science Probability
  1. The term 'large' is relative and depends on what you are comparing it to.
  2. The Law of Large Numbers states that sample averages converge to the true expected value as the number of samples increases.
  3. The speed of convergence in the Law of Large Numbers depends on the variance of the sample, with higher variance leading to slower convergence.

Simple Kelly betting in prediction markets

Unstabler Ontology • 0 implied HN points • 06 Mar 24
💰 Finance Betting Prediction Markets Probability
  1. Kelly betting is a strategy in gambling that maximizes money growth by betting a fixed fraction of one's income each round.
  2. In prediction markets, the optimal Kelly betting strategy involves spending a portion of money on contracts based on the subjective probabilities of outcomes.
  3. The simple Kelly betting rule can be equivalent to the original Kelly rule in cases with two outcomes, providing a more intuitive understanding of betting strategies.

Basics of Probability and Stats - The inequalities that lead to LLN and CLT - PS01

Arkid’s Newsletter • 0 implied HN points • 09 May 23
🚌 Education Mathematics Statistics Machine Learning Artificial Intelligence Probability
  1. The Markov Inequality helps predict unlikely extreme events based on distribution info
  2. The Chebyshev Inequality shows that a small variance means a random variable is close to the mean
  3. The Weak Law of Large Numbers and Central Limit Theorem are essential for understanding probability and statistics in ML