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Probability — A Grammar for Uncertainty

a grammar for uncertainty

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a grammar for uncertainty Key sections include: PROBABILITY; A gambler's letter sparked a science.; Three rules. The whole edifice.; Probability bends to evidence.; How a rational mind absorbs evidence.; Switch and win two-thirds of the time.; Probability times payoff, summed.; Averages of anything tend to normal.; Some distributions don't tame.; Two answers, both useful..

Key sections

  • 01PROBABILITY
  • 02A gambler's letter sparked a science.
  • 03Three rules. The whole edifice.
  • 04Probability bends to evidence.
  • 05How a rational mind absorbs evidence.
  • 06Switch and win two-thirds of the time.
  • 07Probability times payoff, summed.
  • 08Averages of anything tend to normal.
  • 09Some distributions don't tame.
  • 10Two answers, both useful.
  • 11The future depends only on now.
  • 12A grammar that runs the modern world.
  • 13Where to go next.

Topics covered

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Slide outline
  1. 01PROBABILITY
  2. 02A gambler's letter sparked a science.
  3. 03Three rules. The whole edifice.
  4. 04Probability bends to evidence.
  5. 05How a rational mind absorbs evidence.
  6. 06Switch and win two-thirds of the time.
  7. 07Probability times payoff, summed.
  8. 08Averages of anything tend to normal.
  9. 09Some distributions don't tame.
  10. 10Two answers, both useful.
  11. 11The future depends only on now.
  12. 12A grammar that runs the modern world.
  13. 13Where to go next.
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Presentation Transcript

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Slide 01

PROBABILITY

  • ♠
  • ♥
  • A THIRTEEN-SLIDE DECK
  • a grammar for uncertainty
  • ♠♥♦♣
  • FROM DICE GAMES TO BAYESIAN INFERENCE
Slide 02

A gambler's letter sparked a science.

  • ORIGIN — 1654
  • ♦
  • The Birth of a Field
  • In 1654, the chevalier de Méré — a French nobleman with expensive habits — asked Blaise Pascal a curious question: how should the stakes be divided if a dice game is interrupted before it ends?
  • Pascal wrote to Pierre de Fermat. Their correspondence on the "problem of points" — just a few letters across a summer — founded probability theory.
  • "Monsieur, your reasoning concerning the division of stakes is most ingenious…"
  • Pascal & Fermat, summer of 1654
Slide 03

Three rules. The whole edifice.

  • FOUNDATIONS — 1933
  • ♣
  • Kolmogorov's Axioms
  • NON-NEGATIVITY
  • P(A) ≥ 0
  • No event has negative probability. Uncertainty has a floor.
  • NORMALIZATION
  • P(Ω) = 1
  • Something must happen. The total probability of everything is one.
  • ADDITIVITY
  • P(A∪B) = P(A) + P(B)
  • For disjoint events, probabilities add. (Generalizes to countably many.)
  • From these three lines — published by Andrey Kolmogorov in 1933 — the entire mathematical theory follows.
Slide 04

Probability bends to evidence.

  • CONDITIONAL PROBABILITY
  • ♥
  • Given What We Know
  • Once we learn that B has occurred, the universe of possibilities shrinks. Conditional probability rescales what's left:
  • P(A | B) = P(A∩B) / P(B)
  • Two events are independent when conditioning on one tells you nothing about the other:
  • P(A∩B) = P(A) · P(B)
Slide 05

How a rational mind absorbs evidence.

  • BAYES' RULE — 1763
  • ♥
  • The Great Updater
  • P(H | D) =
  • P(D | H) · P(H)
  • P(D)
  • PRIOR
  • P(H) — what you believed before seeing data.
  • LIKELIHOOD
  • P(D|H) — how well the hypothesis explains the data.
  • POSTERIOR
  • P(H|D) — what you believe after seeing the data.
  • "When the facts change, I change my mind. What do you do, sir?"— ATTRIB. KEYNES · A BAYESIAN MAXIM
Slide 06

Switch and win two-thirds of the time.

  • MONTY HALL
  • ♣
  • A Famous Paradox
  • Three doors. One hides a car; two hide goats. You pick door 1. The host (who knows) opens door 3, revealing a goat. Now — switch or stay?
  • Intuition says 50/50. Intuition is wrong.
  • Your initial pick had a 1/3 chance of being right. Nothing changes that. So the other unopened door must carry the remaining 2/3.
  • Vos Savant published the answer in 1990. Hundreds of PhDs wrote in to say she was wrong. She wasn't.
  • 1YOUR PICK
  • 2SWITCH HERE
  • 3🐐
  • Stay: P(win) = 1/3 · Switch: P(win) = 2/3
  • ▶ SEE IT EXPLAINED ON YOUTUBE
Slide 07

Probability times payoff, summed.

  • EXPECTED VALUE
  • ♠
  • Why The House Always Wins
  • E[X] = ∑ x · P(X = x)
  • An American roulette wheel has 38 slots. Bet $1 on red; you win $1 with probability 18/38, lose $1 with probability 20/38.
  • E[gain] = (18/38)(+1) + (20/38)(−1) = −$0.0526
  • Five cents per dollar. Per spin. Forever. The arithmetic of the casino is patient and unforgiving.
  • 5.26%
  • House edge, American roulette
  • ∞
  • Player's expected loss, played long enough
  • ♥♣♦♠
Slide 08

Averages of anything tend to normal.

  • CENTRAL LIMIT THEOREM
  • ♦
  • The Bell Curve's Secret
  • Take any distribution — uniform, skewed, weird. Sample n independent draws, average them. Repeat.
  • As n grows large, the distribution of those averages approaches a Gaussian, with mean μ and standard deviation σ/√n.
  • This is why the bell curve appears everywhere: heights, test scores, measurement errors. The world is full of sums, and sums become normal.
  • "There can scarcely be anything more remarkable than this law."— FRANCIS GALTON, 1889
Slide 09

Some distributions don't tame.

  • FAT TAILS
  • ♥
  • When the CLT Breaks
  • Power laws obey P(X > x) ∼ x−α. Their tails decay slowly. The variance can be infinite; the central limit theorem fails.
  • Wealth, city sizes, war casualties, financial returns, earthquakes — they don't cluster around a "typical" value. A single observation can dwarf the sum of all the others.
  • "In Mediocristan, you average. In Extremistan, the largest sample dominates."— NASSIM TALEB, THE BLACK SWAN
  • Pareto, Lévy, Cauchy — black swans live here.
Slide 10

Two answers, both useful.

  • TWO SCHOOLS
  • ♠
  • What Is Probability?
  • FREQUENTIST
  • Probability is a long-run frequency. P(heads) = 0.5 means: if you flipped forever, half would be heads.
  • Parameters are fixed but unknown
  • Confidence intervals, p-values
  • Fisher, Neyman, Pearson
  • BAYESIAN
  • Probability is a degree of belief. It applies to one-shot events: "P(rain tomorrow) = 0.7" makes sense.
  • Parameters have distributions
  • Posteriors, credible intervals
  • Bayes, Laplace, Jeffreys, Jaynes
  • Modern practice quietly raids both camps. Cheap computation has tilted the field toward Bayes.
  • ▶ BAYES' THEOREM ON YOUTUBE
Slide 11

The future depends only on now.

  • MARKOV CHAINS
  • ♣
  • Memoryless Processes
  • A Markov chain is a sequence of states where the next state depends only on the present — not the path that brought you there.
  • P(Xn+1 | Xn, …, X0) = P(Xn+1 | Xn)
  • PageRank — Google's original algorithm: a random surfer's stationary distribution.
  • MCMC — Markov Chain Monte Carlo: sampling from impossible distributions.
  • Hidden Markov Models — speech recognition, gene-finding, finance.
  • LLMs — transformers are not Markov, but generate token-by-token in a similar spirit.
  • Three states, transition probabilities. The past forgotten given the present.
Slide 12

A grammar that runs the modern world.

  • MODERN APPLICATIONS
  • ♦
  • Probability At Work Today
  • MEDICINE
  • Bayesian diagnosis combines prior prevalence with test sensitivity. A "positive" rare-disease test is usually still a false alarm — the math forces this on us.
  • INSURANCE
  • Actuaries price risk by integrating distributions of loss. The whole industry is a bet that the central limit theorem holds across enough policies.
  • MACHINE LEARNING
  • Modern ML is dressed-up probabilistic inference: cross-entropy loss, softmax outputs, Bayesian neural nets, diffusion models, RL.
  • SPORTS ANALYTICS
  • Win-probability models, Elo ratings, Moneyball-style player valuation. Every fourth-down decision now has a posterior.
  • CRYPTOGRAPHY
  • Randomness is the raw material of secrecy. Probabilistic primitives underwrite every secure connection on the internet.
  • SEARCH & AI
  • PageRank, language models, recommender systems, autonomous driving — all are inference engines over uncertain worlds.
Slide 13

Where to go next.

  • FURTHER READING
  • ♥
  • ♣
  • Closing Hand
  • BOOKS
  • The Drunkard's Walk — Leonard Mlodinow
  • Probability Theory: The Logic of Science — E.T. Jaynes
  • The Signal and the Noise — Nate Silver
  • The Black Swan — Nassim Taleb
  • How to Solve It — G. Pólya (a classic)
  • Bayesian Data Analysis — Gelman et al.
  • YOUTUBE & ONLINE
  • Monty Hall problem — explained
  • Bayes' theorem — explained
  • 3Blue1Brown — "Bayes theorem" series
  • Veritasium — "The Math of Streaks"
  • StatQuest — bite-sized statistics
  • MIT OCW 6.041 — Probabilistic Systems
  • "Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means."
  • — BERTRAND RUSSELL
  • ♠♥♦♣
  • FIN · THIRTEEN OF THIRTEEN
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