Overview

Probability is the logic of uncertainty. It quantifies chance. It tells us that even in a random world, there are patterns we can bet on.

Core Idea

The core idea is frequency. If you flip a coin enough times, it will land on heads 50% of the time. Probability is the ratio of desired outcomes to total possible outcomes.

Formal Definition

A number between 0 (Impossible) and 1 (Certain). $P(A) = \frac{\text{Number of ways A can happen}}{\text{Total number of possible outcomes}}$

Intuition

  • The Coin Flip: The simplest model. 50/50.
  • The Weather: “30% chance of rain” means in 3 out of 10 days with these conditions, it rained.
  • Independence: The coin has no memory. If you get 10 heads in a row, the chance of the next one being heads is still 50%. (Gambler’s Fallacy).

Examples

  • Bayes’ Theorem: Updating your beliefs based on new evidence. (If a test is 99% accurate, but the disease is rare, a positive result might still be false).
  • Law of Large Numbers: The more you play, the closer you get to the expected average. The casino always wins in the long run.

Common Misconceptions

  • Misconception: “One in a million” means it won’t happen.
    • Correction: In a country of 300 million, “one in a million” events happen 300 times a day.
  • Misconception: Randomness looks random.
    • Correction: True randomness often has clumps (like the iPod shuffle problem). We expect “evenly spaced,” which is not random.

Applications

  • Finance: Risk management.
  • Insurance: Calculating premiums based on life expectancy.
  • AI: Machine learning is just fancy probability.

Criticism and Limitations

  • Black Swans: Probability models often fail to predict extreme, rare events (like the 2008 crash) because they assume a Normal Distribution.

Further Reading

  • The Drunkard’s Walk by Leonard Mlodinow
  • Thinking, Fast and Slow by Daniel Kahneman